Vaccination after neuter?

I rescued a stray cat about a week ago. He is very friendly and comfortable inside. I took him to a clinic for a wellness exam on Thursday, where they told us he was perfectly healthy and gave him a rabies vaccine. Then we got him neutered yesterday.

Today, I realized he was never given a FVRCP vaccine. He had a blood test done, which again, they said was healthy, so I guess that's why they didn't give it to him?

I am a first time cat owner, and he lives alone with one dog currently. But soon, he is living with my aunt temporarily who has three dogs.

My question is can I take him to a clinic to get the FVRCP vaccine before he goes and lives with my aunt? He wouldn't be around any other cats, but I would be worried about him getting/giving anything from/to the dogs. Is he okay to get that shot even though he is still recovering from his neuter? Or is he okay to not have the shot until he is fully healed, since he won't be around other cats?

We don't have an established vet yet, since I rescued him kind of impulsively. He's been going to an SPCA clinic for the initial wellness check and neuter. But should I make an appointment with a real vet even though he just had a wellness exam? He could get the shot there as well. I just want to make sure he has someone to go to make sure his neuter healed well, see if he needs the shot, or incase something else comes up in the future.

Any advice is appreciated, as I am a new cat parent.