Field and Forest has this handy chart that shows which species of mushroom can be grown on which species of log:
https://www.fieldforest.net/category/growing-mushrooms-on-logs
This chart shows that several species of mushroom can be grown on mulberry, including oysters, nameko and turkey tail. But it looks like lion's mane and comb tooth are the best. Which is good, since these are two of the best tasting and most beneficial of all mushrooms. Field and Forest has great spawn and they're running a bunch of specials now. Their videos are very informative and explain everything clearly.
Good luck!