It seems safe to assume that Bavaria wasn't an option as Arnulf the Bad controlled this area throughout Henry's reign. The only possible candidates that I can find are Memleben (where Henry died) or Wallhausen (where he married Matilda and his son and future king Otto I was possibly born).
Wherever Henry the Fowler happened to be.
During this period, the kingdom was basically ruled from wherever its king held court. There was no single "ruling seat" per se since the King tended to move around his kingdom a lot, hunting, touring as well as campaigning. The closest is probably the several preferred residences that kings would have had. These took on the appearance of a capital by virtue of housing the royal entourage.
Kings did not just rule from one central place, but also moved around their kingdoms. Charles the Great in later life and Louis the Pious stayed for long periods of time at Aachen, which had something of the functions of a capital. But they did not reside there permanently ... the court was where the king was.
- Reuter, Timothy. Germany in the Early Middle Ages c. 800-1056. Longman, 1991.
In the case of Henry the Fowler, Memleben is probably his favoured residence. Wallhausen, where his son built a palace, is essentially in the same neighbourhood. Thus, the whole region might be considered Henry's "ruling seat", insofar as he had one fixed seat of power.
Memleben on the Unstrut, about ten miles south of Allstedt, was the favourite residence of the German emperors of the Saxon line ... Here, probably, Henry the Fowler was busying himself with his falcons when it was announced to him that he was chosen emperor; and here, too, he breathed his last .
- Köstlin, Julius. Life of Luther. Longmans, Green, 1895.
Neither Regensburg nor Frankfurt can quite be described as a "capital" of the east Frankish kingdom, but - especially taken together with other favoured royal residences in their neighbourhood - they came close to fulfilling such a function.
- Reuter, Timothy. Germany in the Early Middle Ages c. 800-1056. Longman, 1991.
Another notable location was Aachen. This city is often cited as the capital of the Frankish empire, since it was the old seat of Charlemagne. Its significance can be seen in the fact that Otto the Great was crowned there as the successor of Charlemagne. The Kings of Germany would continue to be crowned at Aachen's Imperial Cathedral until the 16th century, though Aachen's actual political status withered much earlier.
Aque or Aix-la-Chapelle, where Charlemagne died in 814, and Otto I was crowned in 936 with great solemnity, continued to be considered as the capital of the empire.
- Koeppen, Adolphus Louis. The World in the Middle Ages: An Historical Geography. 1854.
There was just no ruling seat at all in that time.
This is hinted at in the other answer but can be misread it seems. So "Wherever Henry the Fowler happened to be" should really be understood as: there was no permanent 'seat' and yet, Henry never left his court, as the court was always with him as he travelled, which he did a lot.
That king Henry was born in Memleben but crowned at Fritzlar. But these were just stations in his itinerant kingship with little significance over others. For example Henry IV crossed the entire country within one year:
During the course of a year, impressive distances were passed through. German historians calculate for example, on the basis of royal letters and charters, that Emperor Henry VI and his entourage in 1193 (between January 28 and December 20) traversed more than 4,000 kilometres - crisscrossing the entire German area.
A reconstruction of destinations gives the following chronological route: Regensburg – Würzburg – Speyer – Hagenau – Straßburg – Hagenau – Boppard – Mosbach – Würzburg – Gelnhausen – Koblenz – Worms – Kaiserslautern – Worms – Haßloch – Straßburg – Kaiserslautern – Würzburg – Sinzig – Aachen – Kaiserswerth – Gelnhausen – Frankfurt am Main – and finally Gelnhausen again.
Despite Henry's attachment to Memleben, it can be argued just as well that for example Pfalz Pöhlde was the main Pfalz for him, as this became the most often visited Palace druing Christmas times. (Günther Binding: "Deutsche Königspfalzen: von Karl dem Großen bis Friedrich II. (765 - 1240)", WBG: Darmstadt, 1996, p165.) And equally in Henry's last year big actions as a ruler were a planned but not realised trip to Rome, a meeting of kings in Ivois, back to the Harz mountains were he suffered a stroke, a last court in Erfurt and then he almost retired to die to Memleben, but was buried in Quedlinburg, the town where he wanted to be memorialised. Of the four feasts of Easter that we can attach to a place, 3 were in Quedlinburg, a place of special significance for the Ottones around 900 (Babette Ludowici: "Quedlinburg vor den Ottonen: Versuch einer frühen Topographie der Macht", in: "Frühmittelalterliche Studien." Vol 49, 2015, p91–104.)
It is said the Charlemagne preferred Aachen as his 'seat'. But while it is true that he liked the place and built a Königspfalz there, he really did so only late in his life when he was not travelling administratively so much anymore or even early when he was quite much on campaign. It seems that this was just an effect: after being in the saddle for decades and then opting for sedentary proximity to warm springs when not many challenged his rule. This challenging the power however was again a constant for his successors.
These Pfalzen were often built within 30 km distance and numbered at the end into the 360s. As they were meant to supply the kingly court for a longer stay than the other places suitable for short visits: monasteries or cities, who were put under heavy strain for such a visit.
Only later during after the 13th century did this 'horsing around' shrink and fossilise to very few places, mainly between Nürnberg, Regensburg, Frankfurt and Ulm. (Hans Conrad Peyer: "Das Reisekönigtum im Mittelalter", Vierteljahrschrift für Sozial- und Wirtschaftsgeschichte, Vol 51, 1964. / Wilhelm Berges: "Das Reich ohne Hauptstadt.", in: "Das Hauptstadtproblem in der Geschichte". Tübingen 1952.)
To get a glimpse of the distribution of all those Pfalzen, there a few maps available at Repertorium der deutschen Königspfalzen.
That would mean that a central assumption from the question has to be addressed differently: Bavaria was a region of special interest for the king, so he would have to visit quite frequently. Denying him this right would mean open rebellion and result in war. Which it (almost) did. As Henry went after Arnulf and after a short time Arnulf submitted to the king and opened the gates of besieged Regensburg for him. Afterwards he was called "friend of the king" and followed him into war against Bohemia and Hungary. (Compare the partisan depiction in Fragmentum de Arnulfo duce Bavariae).
All those squares on the following map add to the cities and monasteries called to duty during a visitation:
1: 2: 3: 4:
1: location of Pfalzen in Bavaria projected onto modern map, // 2: Die bayerischen Teilherzogtümer und die Oberpfalz um 1350 from: Gertrud Diepolder: "Bayerischer Geschichtsatlas", Max Spindler (Ed), München 1969, p. 20. // 3: And an old map from 1634 map of the Duchy of Bavaria (click on "Zur historischen Karte" im DigiTool-Viewer) // 4: Karte des Herzogtums Bayern im 10. Jahrhundert
The successor Otto's royal household devoured daily 1000 pigs and sheep, 10 bales of wine, 10 bales of beer, 1000 malts of grain, 8 cattle, plus chickens, piglets, fish, eggs, vegetables and much more (according to the contemporary source "Annalista Saxo").
There were no fixed residences. Rather, the ruler moved from court camp to court camp. Towards the food, the taxes. The court moved around because the farmers' contributions in kind and the money and tax contributions of the cities had to be collected in various places. Also the homage of the estates and the jurisdiction had to take place locally. In the late Middle Ages and early modern times, however, the further development of the urban system and the rise of the early capitalist monetary economy created the conditions for residential cities such as Prague, Vienna and Dresden. (WP: Residenzstadt)
As @Pieter Geerkens commented, there was also a quite general effect to be observed:
insufficiency of the commercial and travel networks, and even of city size, made permanent seats of government impracticable. There is a reason why London, Paris, Madrid, and Vienna, the early Western European permanent seats, are all on substantial rivers.