I am a Black American with the expected recent admixture of West/East African and Western European ancestry, but I also have some more distant South Asian ancestry (estimated to be most recently from about the 1700s) and a maternal haplogroup originating and most commonly found in Bangladesh/Nepal/Myanmar. Of course, there is the possibility that the ancestry came by way of my European ancestors mixing with South Asian people during early colonization there, but I haven't found any evidence of non-Black female ancestors in my family, so I was wondering if it were possible that a South Asian ancestor migrated to Africa and her descendants were later enslaved and sent to the Americas?
No documentation in the sense of historical i.e. written materials suggesting Asian migration to anyplace in Africa before modern times, with one notable exception-- Madagascar. Malagasy genetic signatures are quite distinct so if that were the source of the signal, you'd likely have some indication of that heritage.
We do have written materials showing all kinds of trading in the Indian Ocean, and visits by Near Eastern, South Asian and even East Asian merchants and sailors to East Africa. These materials don't mention any migration or permanent colony, and the movement of people is mostly _from_ rather than _to_ Africa: a very steady slave trade from Zanzibar to the Persian Gulf (African slaves launch what is perhaps history's largest slave revolt, the Zanj Rebellion in the 9th century in what's now southern Iraq).
There are communities in what are now India and Pakistan that have traditions of descent from African slaves, and genetics confirms that these people (variously called "Siddi" or "Makran") do have this heritage.
In terms of likelihood, the most likely source for South Asian genes in someone of African descent would be from South Asians arriving in the 19th century-- although you've got a date of "the 1700s", if I were placing a bet, the most likely scenario is that that date is just a bit too early, and that the genetic signal comes from someone arriving a generation or two later. The British develop a system of Indian indentured labor after the end of the slave trade, and you have large Indian populations in places like Natal.
One other possibility would be travel within the Muslim world-- Africans made the hajj, most famously the Malian King Mansa Musa. Its easy enough to see how an Arab or Berber genetic signal might be found, but to find a South Asian heritage from places that were not significantly Muslim is more of a puzzle.
To know more, it would be helpful to know just _which_ South Asian genes you might have-- and this is a question for a population geneticist rather than a historian; a very hard look at a drop of your blood is going to be the most informative data you'll get. If your South Asian genes can be narrowed down a bit in origin, a historian of the British indenture system might be able to give you some idea if folks with that particular genetics were likely to have been among the Indians transported to Africa.
[Edit] -- the most likely source of a genetic signal in an African American (as opposed to an African) would be through the Caribbean; large Indian communities in places like Trinidad, had contact with New Orleans in particular . . . that would be the most likely, though again it would have to be just a bit later than the 1700s.
Sources
Genomic landscape of human diversity across Madagascar.
Indian Siddis: African Descendants with Indian Admixture
Unraveling the Population History of Indian Siddis
A New System of Slavery: The Export of Indian Labour Overseas 1820-1920