Hi, I don't know much about Bassano's claims, but broadly speaking the Shakespearean authorship question is a big old red herring. No contemporary account anywhere casts doubt on his authorship. We also have multiple contemporary quotes about Shakespeare, see for example this poem by his friend (and fellow) playwright Ben Jonson just after Shakespeare died. Conspiracy theorists will say e.g., that either Jonson was in on the conspiracy or that Jonson actually wrote Shakespeare himself, but conspiracy theorists don't evaluate evidence like the rest of us.

The authorship question seems to have originally stem from snobbish people deciding that Shakespeare was not educated enough to have written his plays.

This is a fantastic answer by a flaired /r/AskHistorians user who knows WAAAAY more about this than me.

It is also worth noting that, according to wiki at least, there is no consensus that Amelia Bassano was actually 'black'.