Disclaimer: I am not a vet and this is not veterinary advice. However I live in an area with a very large stray cat population and I’ve helped probably 30 cats get fixed over the years, so I have a lot of experience with situations like yours.
First and most importantly, please separate Noir and Paxton until you are able to get Nior tested. FIV and FeLV are both very common in strays. These typically (but not exclusively) spread through fighting, and intact males are much more likely to engage in fighting behaviors so the chances that Noir is already positive for one or for both are relatively high — especially since he had a visible injury when you found him. If you’re able, keep him isolated (i.e. in a dedicated room with the door kept closed) and with his own litterbox since FeLV can spread via shared litter.
As far as getting him fixed is concerned, you should be fine doing it as soon as possible. The surgery for males is not very invasive and the recovery time is much shorter than with females. If Noir is moving around on his own, eating and using the litterbox without help, etc, he’s probably healthy enough to be fixed.
When booking his surgery it’s fine to go through a traditional vet, but you can also look for a local “TNR” program (trap, neuter, release). Most shelters run these themselves or can put you in contact with a local organization that does. Typically this will be several hundred dollars cheaper than a traditional vet, and they can test for all of various diseases and parasites that a stray could reasonably have. They can also give all of the necessary vaccines as part of the procedure, and provide you with any meds required for their aftercare. Usually TNR programs will have you drop the cat off in the morning and then pick them up the next day so they can recover with you at home. This is all the more reason to set up a dedicated space for Noir since he’ll probably want to be left alone after his surgery.
Finally, on a personal note, thank you so much for volunteering your time, money and personal space like this. Most people don’t realize just how bad the stray situation is in many places, and even fewer are willing to help out. Just know that there are people who sincerely appreciate what you’re doing — and I know Noir will too!
I know it sucks, but it’s really the only thing you can do in this situation. FeLV is no joke, and it’s rare that an infected cat lives longer than 2 years, even with proper care. Noir is in the highest risk group for FeLV, and if he does have it being in direct contact with other cats greatly increases the chance of it spreading — especially if he’s sharing a litterbox.
The temporary inconvenience to one cat could end up being a live-saving intervention to the other.