Ok, after continuing my research on the subject and digging deeper, I found a source from Henry L. Savage, I didn't have access previously to, that goes into a little more detail on the relationship of Enguerrand VII de Coucy with his 2 lieges. Enguerrand was sent along with other French nobles to England as hostages for the ransom of the King:
In 1363 Coucy went to England as hostage for the ransom of his king, who had
been captured at Poitiers in 1356...
There he gained favor with Edward III and was married to his daughter Isabella.
But at London, Eltham, and Windsor he probably met and conversed
with some of the most brilliant soldiers of whom England boasts, the Black Prince,
Chandos, Audley, Burghersh, and Loryng. He could not have failed to have prof-
ited by such association. His marriage to Isabella, eldest daughter of the English
monarch, in 1365, his assumption of the title of Earl of Bedford iLa the succeeding
year, the grant to him of lands that had belonged to him in Kendal and Lanca-
shire, led to consequences that were important for his military career. If one
wonders why the English monarch should have loaded a hostage with such favors,
the answer is to be found in the fact that the Sire de Coucy, as perhaps the most
powerful baron in Northern France, could have been very useful to the monarch's
schemes, if he had been won over to the English cause. And I believe that a second
motive for Edward's favors is to be found in the high respect he entertained for
the Sire de Coucy's military ability.
When hostilities resumed between France and England, Enguerrand chose to support neither.
In 1369 war again broke out between Coucy's liege lord and his royal father-in-
law. To which side should, or could, he adhere? He hit upon a solution that was
as tactful towards his superiors as it was honorable to himself. Renouncing a
theatre of activity that would have provided full opportunity for the exercise of
his powers and the enhancement of his prestige, he withdrew to Italy to evince his
loyalty to the church by serving in the papal forces operating against the Visconti, tyrants of Milan.
Upon his return from this expedition, he again found a way to avoid direct involvement in the conflict.
King Charles V now (quite tactfully) suggested that, following the example set by Du Guesclin in his Spanish expedition, Coucy should rid the country of many of the 'thugs' enrolled in the free-companies. Enguerrand's oft denied claim of the allodial possessions in Alsace and Switzerland belonging to his mother by right of her being an Austrian princess, furnished a convenient excuse for the enrollment of numerous 'thugs' to enforce it against the Hapsburgs.
Additionally respect and tactful approach was shown from the English as well towards de Coucy as told in this source:
Enguerrand returned to Coucy in 1374 where he found his lands in good order. Robert Knollys who had traveled through Picardy, had been ordered by Edward III to leave the lands of Enguerrand intact. On 4th August 1374 Charles paid Enguerrand 6,000 Francs and in November 1374 asked him to become the Marshal of France, but Enguerrand declined.
Finally, he was freed from his dual obligations with the death of Edward.
By the early months of 1376, Coucy was back in France, and in the next year there occurred an event which allowed him to play a much more active role in affairs than he had done. The death of his royal father-in-law removed one to whom Coucy was under deep obligations for favors and for a solicitous regard for his interests and prestige. He renounced his English honors, and returned his Garter, and, since she seemed to wish it, his wife as well, to the nation that had given him all three. He was now free to devote his interests to his suzerain.
Enguerrand stayed neutral in this conflict as neither of his lieges really pressured him in joining the war. He was a skilled diplomat as well as a commander and it seems his neutrality was indeed thanks to personal charm:
Few of his countrymen have practised diplomacy as successfully as Enguerrand de Coucy. It was not that he was false or evasive, or prolific in promises - he could be canny enough, when there was need of canniness. His ability came from a rare personal charm. All his life he was praised for his graciousness and generosity, and thrice at important moments of his career he refused promotions or honors that he might have obtained, out of consideration for others - or for the preservation of his own self-respect. His desire to conciliate and appease ruffled
feelings, the knowledge of men which enabled him to clothe a suggestion in its most enticing form, his reluctance to bring matters to an issue - until he had to do so - made him one to whom kings, regents, and trained civil administrators turned in the last resort with the request that he take the matter in hand.
I haven't been able to find his feudal contract, if such thing even exists. But, to answer my original, more general question about how a vassal could deal with dual allegiances, I think it is safe to say that in the Middle-Ages personal ability mattered the most when dealing with sovereigns.