A Google Books search for "professional bias" for the period 1700–1800 yields four legitimate matches—all of them connected to religion. From Richard Watson, An Apology for Christianity in a series of Letters, Addressed to Edward Gibbon, Esq (1777):
I beg pardon for styling their [the Deists'] reasoning, prejudice ; I have no design to give offence by that word ; they may, with equal right, throw the same imputation upon mine ; and I think it just as illiberal in Divines, to attribute the scepticism of every Deist to wilful infidelity; as it is in the Deist, to refer the faith of every Divine to professional bias.
Bishop Watson's response to Gibbon is cited in Letter 44 of Richard Sullivan, A View of Nature, in Letters to a Traveller Among the Alps (1794):
A latent, and even involuntary scepticism, certainly adheres to some characters. And therefore, it is illiberal in the advocates of religion, to attribute the scepticism of every Deist to perverse infidelity ; as it is in the Deists, to refer the faith of every Christian to professional bias.* This particular bent we can neither comprehend, nor estimate.
As Hot Licks notes in a comment below, this is by no means an independent occurrence of "professional bias," but rather a restatement of the previous instance, with Christian substituted for Divine.
From a letter to the Philological Society of London by N. N. on March 7, 1787, in The European Magazine (March 1787):
We have a hint also of "the number and ability of unbelievers." I will not class the Reviewer with those Free-thinkers, as they call themselves, who are mere slaves to the opinion of others ; though I suspect him to have very little knowledge of the facts or answers in defence of Christianity. With those, however, who disbelieve, not from any reason they themselves can give, but because some acquaintance of theirs, of whom they have a good opinion, or some celebrated writer, as Voltaire, Hume, disbelieved, we may argue in their own way, and confront them with names and authority, I trust, superior to any they can produce. ... To say nothing therefore of the bulk of the community, high and low, rich and poor, learned and illiterate, which or so many ages have believed in the Gospel, let us only urge the names of Mede, Cudworth, Barrow, Clarke, Jortin, ; of Leland, Taylor, Lardner ; of Le Clerc, Limborch, Mosheim ; men who spent whole lives in the study of Christianity, and manifested as much freedom and acuteness in their researches, as are to be found in any science whatever. Let us add the authority of Bacon, Grotius, Locke, Newton, Hartley, men who were under no professional bias, and did not take their religion upon trust, but each of them spent many years in inquiries into it, and rose up from the inquiry fully and firmly persuaded of its truth.
And from The Parliamentary Register (April 28, 1795):
The Bishop of ROCHESTER disclaimed having any professional bias : he said, in all great bodies of men there were some undeserving objects, but it would be unjust to punish the worthy on their account.
As two of the four eighteenth-century instances of "professional bias" in the Google Books search results are from bishops, a third quotes one of the first two, and the fourth distinguishes between "men who spent whole lives in the study of Christianity" and men who merely "spent many years in inquiries into it" (arguing that the latter did not have a professional bias), it seems clear that prior to the 1800s the term was understood to refer to religious profession.
The earliest nonreligious (or religion-neutral) instance of "professional bias" in a Google Books search appears in a review of John Fuller, M.D., The History of Berwick upon Tweed, in The Monthly Review (October 1800):
This design [to improve "the present state of agriculture and commerce of his native town" and to propose "the real happiness of the inhabitants"] is no doubt truly benevolent and patriotic : but surely it was not necessary, in order to impress on the reader's mind the importance and utility of agriculture, to give an account of man in a savage state ; nor to present us with various other observations which here occur, and which seem to originate in the professional bias of the author's ideas.
Since the author is a medical doctor and not a divine, I assume that his professional bias is in the direction of modern medicine. The review doesn't mention religion at all.
This rather limited record supports the idea that "professional bias" began as a term connected to the profession (that is to say, the professing) of Christianity and used by various religious and nonreligious writers, and that from there it expanded to include professional occupations or livelihoods within which a particular viewpoint or presumption or interpretive inclination predominates.