Haha! Cool! I ran a CNET BBS from the Amiga 500. As far as I know, there are no processor requirements for any version of CNET. (though the 3.x series may require a 020+). There were versions, maybe starting as early as the 2.x series that required OS2.04+, though. I don't think any ever required 3.1, though.
You didn't say what ROM you were running in the A500. I'll assume the stock 1.3. I am positive you can run CNETs through the 1.9x family on that. Seems like I used to run a 1.92 or 1.98, or something. My memory is pretty hazy. (We're talking almost 15 years ago!) Looking through the BBS disks here, I have a backup of my friend's old CNET 2.63, and a backup on version 3.05c also, but by that time it was running on an A1200, so that's difficult to say what the requirements were.
If you can't find anyone associated with the CNET anymore (does Ray Akey still come here?), alternatives include
The great Excelsior! BBS (my 2nd favorite to the old-school CNETs), which I believe is now available as a freeware download. (Though I can't find a link ANYWHERE! Incredible!) I'll check the dox on my 4000 later and let you know, if no one else has found it...
The DLG BBS wasn't too bad, IIRC. I have a demo copy here, but never ran or personally knew anyone who did, though I seem to remember calling a few.
Those were the "Big 3" around here, besides the stupid PC TAG and Wildcat! systems.