tl; dr
No, Edward III paid a token tribute of £1,000 in 1333 (in expectation of receiving papal favours in return).
In 1365, the English parliament debated the latest papal demand for tribute. They concluded that John’s original surrender of the realm to the Pope had been invalid, since it had lacked the assent of the bishops. From the perspective of the English, that meant that England had never actually been a Papal fief in the first place.
Since no English king paid tribute to the Pope as vassal after that date, 1365 would seem to be the year that England ceased to be a Papal fief.
Background
When King John surrendered England to the Papacy in 1213, he also agreed to pay an annual tribute to the pope of 1,000 marks (1 mark was worth 13 shillings and 4 pence). This tribute was never paid regularly, although it was paid from time-to-time into the fourteenth century.
"... of our own good and spontaneous will and on the general advice of our barons we offer and freely yield to God, and to SS Peter and Paul His apostles, and to the Holy Roman Church our mother, and to our lord Pope Innocent III and his catholic successors, the whole kingdom of England and the whole kingdom of Ireland with all their rights and appurtenances ..."
"... As a token of this our perpetual offering and concession we will and decree that out of the proper and special revenues of our said kingdoms, in lieu of all service and payment which we should render for them, the Roman church is to receive annually, without prejudice to the payment of Peter's pence, one thousand marks sterling five hundred at the feast of Michael and five hundred at Easter that is, seven hundred for the kingdom of England and tree hundred for the kingdom of Ireland, subject to the maintenance for us and our heirs of our jurisdiction, privileges, and regalities."
- Concession of the Kingdom to the Pope made by King John before Pandulf, the Papal legate at Dover on 15 May 1213, and renewed at London, before Nicholas, Bishop of Tusculum, on 3 October 1213
Note that this explicitly excluded the payment of Peter's Pence.
In return, the Pope issued a Papal bull placing England under his protection.
It was on the grounds that England was a Papal fief that Pope Innocent III issued a Papal bull on 24 August 1215 declaring Magna Carta to be null-and-void. The reasoning was that the charter would have violated his rights as feudal lord.
King John's son, Henry III, enjoyed close relations with the papacy throughout his reign, with Papal legates at his court (like Pandulf Verraccio) having right of veto on many matters.
However, as you note, John's grandson, King Edward I, and great-grandson, King Edward II, did not enjoy such cordial relations with the Pope. As you say, in part this was about the king's right to tax the English church, and also over their continuing wars in Scotland. However, matters were certainly not helped by the Pope's perceived (and perhaps unsurprising, given the Papal exile in Avignon) partisan support for the kings of France in dealings between the two kingdoms.
Payment of Tribute
It seems that tribute was paid (at least intermittently) during the reign of Henry III.
However, as a result of the increasing distance between the English kings and the Papacy during the reigns of Edward I and Edward II, payment became less frequent, and no tribute at all was paid between 1300 and 1330.
Edward III paid tribute of £1000 in 1333, and that is the last payment for which have a record (although we do have copies of demands for payment from the Pope).
The parliamentary debate in 1365 was prompted by a Papal demand for the arrears of tribute that remained unpaid. As the Rev. M.W. Patterson put it in his 1929 History of the Church of England:
In the year 1365 the Pope was injudicious enough to demand the arrears of the tribute promised by King John for himself and his successors. This claim was emphatically rejected by Parliament, and the papal suzerainty renounced.
Since no English king paid tribute to the Pope as vassal after that date, 1365 would seem to be the year that England ceased to be a Papal fief.
Rejection by Parliament
As I understand the argument, Parliament acknowledged that King John was free to surrender the kingship (i.e. abdicate), but he could not change the succession, and so 'bring his realm under the subjection of another', without the approval of the barons and the bishops (the Councils that were the precursors of Parliament). They also noted that John had surrendered the kingship under duress.
While the Pope may have been head of the church, he had no formal position in the Councils of England. Since John didn't have the approval of the bishops, when he surrendered the kingship, Parliament effectively asserted that his infant son (Henry III) had automatically become king, and not the Pope.
The following extract from Cobbett's Parliamentary History of England describes how Edward III put the Pope's demand to Parliament, and records their response:
... After which both houses proceeded to nominate receivers and tryers of petitions as usual, and adjourned to the next day, when the chan. in the presence of the king, lords, and commons, spoke again and told them, "that he had the day before informed them in general, of the occasion of their meeting, and that now they should know it more particularly; the king having a matter of great importance to communicate to them. His maj. had lately received notice, that the pope, in consideration of the homage which John k. of England, had formerly paid to the see of Rome, and of the tribute by him granted to the said see, intended by process to cite his maj. to appear at his court, at Avignon, to answer for his defaults, in not performing what the said king, his predecessor, had so undertaken for him and his heirs, kings of England. Whereupon, the king required the advice of his parl, what course he had best take if any such process should come out against him." The bps. lords and commons, desired until the following day, to give in their answer; when, being again assembled, after full deliberation, they declared as follows, “that neither king John nor any other king could bring himself, his realm and people, under such subjection, without their assent; and if it was done, it was without consent of parl, and contrary to his coronation oath; that he was notoriously compelled to it by the necessity of his affairs and the inquity [sic] of the times; wherefore the said estates enacted, that in case the pope should attempt any thing by process, or any other way, to constrain the king and his subjects, to perform what he says he lays claim to, in this respect, they would resist and withstand him to the utmost of their power.”
This parl, continued to sit till the 11th of May ...
Payment of Peter's Pence was also suspended for a time under Edward III in the 1520s. Wikipedia notes that:
In 1366 and for some years after, it was refused on the grounds of the pope's obstinacy.
Payment would also be withheld by later kings as a means of applying pressure to the Pope. However, apart from these interruptions, Peter's Pence continued to be paid by the English Church until it was abolished altogether by the Reformation Parliament in 1534.
Further sources
- Ann Deeley: Papal provision and royal rights of patronage in the early fourteenth century, English Historical Review, 1928
- C. Warren Hollister: King John and the Historians, Journal of British Studies Vol 1, No 1, (Nov 1961), pp. 1-19
- Thomas B. Lenihan: “The English Church Shall be Free”: Roots of the Reformation from William I to Henry VIII, MA thesis, 2011
- Thomas W. Smith: 'The Development of Papal Provisions in Medieval Europe', History Compass, 13 (2015): pp 110-121
- Benedict Wiedemann: Papal Overlordship and Protectio of the King, c.1000-1300, PhD thesis, 2017