Like you mentioned in the OP, a number of Albanian intellectuals toyed with the idea of a mass conversion in order to solicit further western support. Catholicism is not a viable option due to previous tensions and antagonism between Muslim and Catholic tribes in the North. Historically, Protestantism attracted the most attention and I think with an early enough POD you might be able to get something going...here's one idea.
POD: Perceiving that the use of the Greek alphabet might be problematic to Albanian readers, members of the British Foreign Bible Society ask Vangjel Meksi to translate the New Testament into Albanian, but using the Latin Alphabet. Meksi reluctantly agrees and comes up with an alphabet of 36 letters all of which are common in Latin typography. Beginning in 1819 (as per OTL) he takes 2 years to finish the work, in addition to a new grammar book using his new Alphabet.
Without the issue of Hellenization posed by the Meksi translation it is much more popular than OTL, particularly among the Albanian elite. Meksi's Alphabet inadvertently jumpstarts an Albanian cultural birth and spawns not only a subsequent translation of the Old Testament, but poetry and prose. The Ottomans try to crack down on the proliferation of Albanian language works, but consistent support from the British Foreign Bible Society in smuggling Bibles in, keeps the fires burning and paradoxically increases the popularity of the Bible and by extension Protestant Christianity.
Pressure continues to build for years as the Albanians universally adopt the Meksi Alphabet and begin to educate their children illicitly with aid from the BFBS. Rights for Albanian Christians are broached from time to time in negotiations surrounding the Tanzimat but are thrown out due to larger strategic considerations. The number of Albanian protestants steadily grows to about 5%-10% of the population by the time of the Treaty of San Stephano.
Now by 1878 in TTL the Albanian "Cultural Renaissance" has already been going for roughly 50 years despite the best efforts of the Turks. Furthermore it's clearly recognized that the Meksi translation of the Bible has been instrumental in initiating it and maintaining it. Protestant influence, present in OTL, is a dominating force in TTL and it has spread to the North in addition to the south of Albania.
Thus when the League of Prizren meets in 1878 there are dozens of delegates from the South, most of whom are Protestant or at least amicable to Protestantism (In OTL there were exactly 2 delegates from Southern Albania). 50 years of Ottoman suppression of Albanian culture in TTL have soured relations between the two people, and rather than call for a maintenance of the status quo within the Ottoman system, in TTL the Treaty of San Stephano is the last straw. The Albanians enumerate their demands which are more reminiscent of the demands of 1912 in OTL than 1878. In order to hammer home their demands for cultural autonomy, the Albanians toy with the idea of a mass conversion just as they did in OTL, only in TTL there's a lot more cultural weight behind it.
"Where the Sword is, there is faith" is an Albanian proverb, often used to characterize their conversion to Islam. I'd imagine that after 50 years of Ottoman oppression in TTL the Albanians would see that the Ottoman Empire is not the place for Albania. I think that a mass conversion would be more seriously considered in the hopes of gaining the UK as a backer. The Northern Tribes could pull this off, as for them cultural tradition matters far more than faith. Thus in TTL at the League of Prizren, the Northern Albanian tribes all convert to the Albanian Evangelical Church or something and they're joined by a number of liberal beys from the center of the country and the south. The population itself is ambivalent at best. The clerics try to lead a revolt but it doesn't get far as the entire country is swept up in nationalistic fervor. Mosques are torn down or converted into Protestant churches. Calls go out to the UK and America for missionaries and aid in order to cement this conversion.
The entire region hovers on a knife edge as the Ottoman Empire tries to figure out what to do. British support, though not a certainty, comes through and the Ottomans give in to almost all the Albanian demands. It doesn't take very long for Albanian demands to be copied in Macedonia and elsewhere in the Empire rapidly changing the very nature of the Ottoman Empire. Due to it's fundamental role in Albanian independence, Protestantism becomes synonymous with Albanian national identity.
Of course, there's a valid point that the success of the Meksi translation will have massive ripple effects on other missionary activity in the Balkans. That might take some more research.
A slightly more plausible option could be a mass conversion to Bektashism (also considered)...but I'm not sure if Bektashism satisfies the conditions of the OP.
