System, you've already gotten some great advice. This is such a wonderful thing you're doing!!
I did want to straighten out one part of the discussion, though already addressed.... like many others that participate in this part of TCS, we trap feral cats. These are completely unapproachable cats (outside our home), and most of them have only had negative interaction with people. There has never been any sedation involved (other than for the spay or neuter), there has never been any chasing with nets or other devices. They are usually so terrified that they are completey subdued when removed from the crate/trap at the vet. They sit in fear for exam, shots, temperature being taken, etc. So if your outside kitty is ever involved in a fight or is injured or visibly sick in some way and you need to get him to the vet, please bear this in mind. And given your vet's response to "wild" cats, I'd suggest that you may want to consider finding a different vet. You may want to go so far as to start feeding kitty in a crate with the door open. If he's been eating inside the crate for four months, it'll be a lot easier to just close that door than to try trapping the cat if he is ever in need of urgent medical attention.
Also - just wanted to add - we had a multi-year old feral that we've been feeding for quite some time (had him neutered last year). Hubby would spend time outside just sitting while he ate. Feeding him regulary got him used to the sound of food, and it got to the point where he'd come running when he heard him shake the kibble.
Little by little, he would move closer to him while he was eating (I'm talking over weeks/months, not days) - and it got to the point where he'd come running when he heard the food, and he trusted hubby enough to get very close to him. Gary never tried to reach out and pet him. I think he was getting close to the point where he was about to headbump Gary for pets - but he got involved in a very bad fight, had his nose ripped up pretty bad, and we had to get him to the vet. We REALLY wanted him to be adopted out as a pet - it's been a subzero Winter here, and we've already got five rescues living full time in an R.V. so there is no way we could bring an older feral that needs socialization inside! Fortunately, one of the vet techs that is experienced with rescuing decided to adopt him - and that was just a few months ago. He's already a cuddle bug.
Also, just so you know there is hope of socializing this guy, we have an inside cat that was all hiss & vinegar from six weeks of age. Tuxedo now has an amazing bond with my husband. This little ball of arched back, flamed tail and hiss was attacking hubby (who didn't try to approach him, was just pouring food in a bowl) from eight weeks of age. His brothers and sisters all became friendly very quickly - but this little ball of fire attacked him, other cats, new cats to the colony, etc. This went on from eight weeks of age - and for the next nine months. And this was despite being fed about 4 times a day by hubby, being fed warmed chicken broth and warmed milk in subzero weather, being dug out of a groundhog burrow after being trapped in there for two days under three feet of snow. It was March of last year, I think, and Gary went out to feed him. He ran over to Gary, attacked him as he was pouring food (!) - and Gary just sat down on the picnic table bench (in the snow) and cried, he was so frustrated. THEN the little nut walked up to Gary and head bumped him on the ankle. And that was it. He came running for pets after that. He loved being held not too long after that. But this was a cat that we began feeding as a kitten. With an older feral it will likely take longer. The trick is to do things on the cat's time schedule, and to let the cat make the first moves. Believe me - they know who they're caretakers are. Once they know you won't ever try to reach out for them no matter how close they get, that's when the real trust begins. But wait for that headbump before trying to reach out for kitty, and despite the fact that he lives outside, I expect that one day you'll end up with a lap cat. It can happen - but so much of it depends upon you, your patience, and the signals you send to kitty.