Although I have 2 MP 4,1 - 5,1’s I also have Big Sur running quite well on a 3,1.
I used Open Core “The Bassic Way” on the 4,1 - 5,1’s. and Mr. Macintosh Legacy Patcher on the 3,1. It was easier for me. I did see where I should have 1st removed the BT card (I did not & no issues so far,) I see also I should have used the “Big Mac” patcher for the 3,1 ..the Guy did a lot of work on it but I did not think it was as easy as Legacy Patcher..with the Big selection of patches & hacks to choose from, I don’t know which one is best. But let us know how it works on your 3,1 Good Luck. If it works as good for You as it is currently working on my 3,1 you will be happy. Hope I didn’t jinx myself..