Should I try to catch a cat now for a neuter appointment in 2 weeks?

A cat has been hanging around our house since early April. I've never been able to get a good look at him in person, but have caught him on camera multiple times. Have also seen him from afar outside during bitterly cold blizzard conditions. He's not eartipped and is skinny and sprays, and I've seen no evidence of kittens so I'm assuming this is an unneutered male.

Once I realized what was going on, I made him a neuter appointment. I had some medical issues and had to cancel one in May and the earliest one available is coming up on June 24th so I scheduled him for that.

I have a feral cat trap and I bought a large cage and was going to attempt to socialize him using the socialization saves lives method. The reason I'm not just going to release him again is that I have been feeding him in an effort to trap train him and he definitely needs the food, however our next door neighbor has a very aggressive indoor outdoor cat who is constantly picking fights with him. I don't see how I would be able to consistently feed him or provide him shelter in the winter without endangering him. I do have another former feral cat who I was able to mostly socialize, she behaves just like a regular cat except I can't touch or pet her. I had her in a room and didn't use the cage and I think that's why I wasn't successful in fully socializing her so I would like to try that with him.

At any rate, I've been very nervous about missing his neuter appointment and again having to delay this. So I was thinking should I catch him a day early and put the trap inside the cage so that he could hang out in there while waiting for his appointment. Then the last few nights he's been more sporadic showing up. He didn't show up at all the night before last and then tonight he came and sniffed the food and wandered off without eating it. I don't know if he doesn't feel well or is getting better food somewhere else. But now I'm concerned he's going to drift away and stop coming here, or perhaps he's ill and that's caused him to lose his appetite, and it makes me wonder if I should just catch him now and keep him in the large cage for 2 weeks while we wait for the appointment. But particularly if he's unneutered I'm not sure if we're going to make make much socialization progress during that time. Would it be best just to hold off till the surgery time?