We won't know whether Duns Scotus faced any hostility for his Scottishness, but we can't expect the perceptions of nationhood and nationality in the late 13th century that would exist if there was a war today. People living in lowland Scotland near to the modern border in the 1290s did not necessarily think of themselves as Scottish, or loyal to a Scottish 'nation'. Certainly they might well have thought themselves to have had more in common culturally and linguistically with the English that the highland and Gaelic-speaking Scots. As a member of religious orders, Duns Scotus was moreover an 'employee', if you like, of the world's largest pan-national organisation, the Church, which did not distinguish people or judge people based on nationality.
Yes, we do know that King Robert Bruce made it his business to make the the noble landowning classes choose a side, by preventing the cross-border landowning which had happened before, but that did not apply to ecclesiastics, who could take benefices or positions wherever they could find them, and continued to do so until the Reformation.
So, while Duns Scotus's very name emphasizes how his place of origin was well understood by contemporaries, being identified as a Scot did not necessarily mean that Scotus was perceived as automatically a supporter of an independent Scotland contrary to the claim of Edward I, or a supporter of Robert Bruce. Scotus' political loyalties will remain a matter of conjecture, but were more likely to be coloured by local South East Scottish family loyalties than an inherent sense of Scottish nationhood or loyalty to his 'nation' of birth. In fact many much more prominent clerics than Scotus remained loyal to the English 'overlord', Edward I and II, even as late as the 1320s.
Finally, it should be noted that Scotus actually spent much of his later life in Paris and Cologne, from 1302 to his death in 1308. It could be that his move to the continent was associated with a desire to lecture in a more supportive environment for Scots than Oxford. However, Duns Scotus's loyalties may have been more to the Church than any secular power, and he was fired from his position in Paris for a time for supporting the pope in a dispute with the French crown.
Finally, the stage of the Wars of Independence in which Duns Scotus was alive was a period in which Scotland underwent periods of complete conquest when the Wars of Independence appeared to be over. From an English perspective, those limited forces that continued to withstand the English victory were small groups of rebels resisting the legitimate government, not the united Scottish nation at war with England. Being Scottish did not ipso facto mean anti-English.