Big Sur will see and connect to Tiger, but it won’t work the other way. Tiger can’t connect to Big Sur. I use this to get things from Dropbox on Big Sur to my MacOS9 laptop then back to Dropbox with Tiger as a bridge both ways. It’s a bit convoluted, but the files need to live on Dropbox for a variety of reasons.
I connect to Tiger from the MacOS 9 laptop and access a folder I set up there to put things in from Big Sur. I don’t need to actually look Tiger or work on it at all now that it is set up.
edited to add
Once the file(s) are back on Tiger, I simply grab them from Tiger on Big Sur and put back on Dropbox