Well, of course the decision to bring him back for the antibiotic injection is up to you. If he were mine, I would have brought him back, or, you could have a housecall or mobile vet come to your home, after conferring with your vet, to find out the blood results and what your cat's examination revealed.
From your description, it's possible that your cat has an ear infection or a condition called vestibular syndrome, which is related to an ear problem and affects balance and equilibrium, causing him to walk like he's drunk. It also makes a cat dizzy and that can cause nausea, like human motion sickness and that's what made him sick, earlier. Here is more information:
If your vet suspects a brain tumor could be causing your cat's symptoms, I would get a second opinion and perhaps ask for a referral to a veterinary neurologist. I would check out the condition of his ears and the possibility of vestibular syndrome before considering a brain tumor.
If you feel he is not in pain, you might wait until tomorrow morning for him to get the antibiotic injection and/or antibiotic drops for his ear(s), after a thorough examination of them. If he can't walk straight, is not eating, not drinking and not eliminating as usual, I would recommend a visit to an ER vet center tonight, to have him examined and evaluated more thoroughly and then, diagnosed and treated, if needed.
I hope all will be well with your furry boy!
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