That's the though part. There would little in the way of obvious reasons to do so except winning the favor ot the Christians. OTOH, many Muslims would quite vocal against it.
Scenario: SHISHMAN EMPIRE ( as a big fat butterfly blowing away the Ottoman from OTL )
1378 - Bulgaria under
http://en.wikipedia.org/wiki/Ivan_Shishman_of_Bulgaria adopts Islam. Only the elite. One "Bulgarian Islam" the same way the
http://en.wikipedia.org/wiki/Bogomilism was shaped as reaction to the compulsory introduction of the Byzantine-centric Christianity. "Bulgarian Islam" without Arabic names, on Cyrilic script, and literaly translation of Quran. Sifu style. Direct continuation of , say,
http://en.wikipedia.org/wiki/Rumi &
http://en.wikipedia.org/wiki/Mevlevi_Order - analogue of the processes which shaped the Iran-centric Islam = slavic-bulgarian centered islamic Mainstream. Add as a spice some
http://en.wikipedia.org/wiki/Bektashi_Order elements evolve in this denomination ( making it more militarized - like the link bektashi-janissaries
http://en.wikipedia.org/wiki/Janissary , thus the Order is the backbone of the state, the Ruling dynasty of Shishmanides, major ethno-converter -- accepting as members of this Royal Order only children, raised as soldiers and considered Childredn of the Tsar ). Gradually during the centuries the "mass-culture" implications of the Shishmanid Order policies evolves into something very alike of
http://en.wikipedia.org/wiki/Bahai consuming and digesting most of the adherents of the Abrahamic ( and not only ) religions, and thus extinguishing the differences, officially regarding the plentiful and various Jewish, Christian, Muslim and other cults as mere practicing nuances of Unified religion. Imagine it as FORCED tolerance, and official strict denial and punishment of any notion of difference!
PoD -- one very different Ivan Shishman - infinitely more cunning, ruthless, cruel and brutal, received a "vision" that this is the only way to save the Bulgarian Kingdom. In 1378 he becomes officially part of the House of Ottoman as brother-in-law ( ref.:
http://en.wikipedia.org/wiki/Kera_Tamara ). Appointed to rule all the Balkan possessions of the Ottomans safe of Eastern Thrace ( where in Adrianople the contemporary Ottoman capital is ).
Very active "BG islam" literature and missionary centers are reopenned and mass producing texts ( like OTL Ohrid and Preslav literaly schools in 9th century ). Thus islam becomes more homey and acceptable for the vast Slavic populations in Eastern Europe.
Secretly Ivan Shishman befriends with Tamerlane, to a degree in 1405 to succeed him on the throne of the Timurid Empire.
In 1389 Ivan Shishman and Tamerlane together take over the Golden Horde and Shishman becomes Tsar ( Emperor ) of the Golden Horde, too. The powerbase of Shishman there are the Volga Bulgarians.
Shisham wins the Ottoman Civil War and heads/supercedes/replaces the Ottomans too in 1402. ( In the period 1387-1402 the Ottoman and many other Anatolian turk states are totally and physically annihilated ).
In 1404 Tamerlane instead of China attacks together with Shishman ( with his vast Balkan, Eastern Europe, Anatolian human resources ) Constantinople and takes it over. Constantinople ( Tsarigrad ) becomes the capital of the Shishman Empire.
The next Shishmanid Tsar is Joseph ( IOTL the Ivan's bastard son, who in our timeline is the Ecumenical Patriarch Joseph II ).
In the next 500 years the Shishman Empire evolves territorially as the Ottoman, Russian, Moghul OTL empires taken together, possessing virtually unlimited human resources for steady "bulgarization" from the expanses of Eastern Europe.
