What type of nail gun is recommended?
I have to say that I am no fan of cordless framer nailers. I have had two different ones, a Porter Cable and a Pasload. Both were gas fired. The PC was a disaster. Had lots of problems with the propane cells seating properly and nails jamming.  Got the Pasload, worked OK, but slow and the smell was so bad that I only used it outdoors.  The gas cells are expensive also.  After two days on a job, I went back to my trusty Bostich 88 and ran the compressor off a gen set. The other thing I hated was that it could not shoot 10d wire collated nails that I use in my other air framers. Those plastic bits flying at your face sting and you have to reload much more often. Absolutely hate plastic collated nails!

To be honest, I have not tried one of the new battery operated models yet. For framing, I like the largest, thickest nail.(3 1/4") and most cordless ones don't shoot the same larger nails I use in my Bostich.

Since your projects are at home and you already have a 135# compressor, I really think you will get a lot more nailer for the $$$ if you stick with a Bostich 88 mag or similar. You will be able to shoot 4d to 10d nails. The unit itself is much lighter, cheaper and very dependable.

For what you pay for a good cordless, you could own a good framer, 16ga finish and 18ga brad gun, thus having the right tool for every nailing job.

I have to say however, a good cordless finish gun is wicked handy and the battery models have plenty of power, but this humble carpenter is not sold on cordless framers.