I had never heard of the Lithuanian Tartary. More info please.
My knowledge of this subject is very limited and I can only redirect to you to
https://en.wikipedia.org/wiki/Lipka_Tatars. The Muslim Tartar aspect is another fascinating aspect of the often ignored Polish-Lithuanian Commonwealth, which is rather extraordinary in having had pagans, Roman Catholics, Eastern Orthodox, Eastern Catholics Lutherans, Calvinists, Jews and Muslims living side by side in relative harmony.
BTW the whole lax attitude to religions denominations makes me smilingly think of this boisterous blasphemic German drinking folk song I recently came across:
https://www.youtube.com/watch?v=IfuQX-CaQg8, whose main theme is: Who would want to be the Pope swimming in indulgence money drinking the best wine when one cannot carouse with the fair sex, and who would want to be the mighty Sultan living in a harem / brothel and abstain from drinking wine as prescribed by the Quran. Much better to be half Pope and half Sultan and drink wine with beautiful women.... :-)