I've been doing it from world view through the manage households interface. I just open it up, move the infant to another household, X out of manage households, then immediately open it back up and add them back. No loading screens inbetween or anything. Probably don't even need to close and reopen the household management thing, but I've been kinda better safe than sorry about it. It does leave the infant lying in a random spot when I load back in (sometimes not even on the lot, idk how it decides where to drop them but sometimes I have to send someone down the road to go collect them), which is annoying. I think someone else was having success just using cheats to add/remove, so I'll probably try that next time I play that household.
Also, if in your case there's no active careers triggering the problem, just a run of vet visits (been there. sometimes those pets just KEEP getting sick) you might be better off buying a few wellness treats from the vet vending machine so that you can heal the cat that way instead, with no actual vet visits required!
Oh, and just so you know, if you only add/remove the infant, milestones will probably remain broken for everyone else in the household. I haven't been paying enough attention to confirm that, but it seems like a reasonable assumption. I've decided that's fine if it means less work for me, cause with 4 active career sims in the house I'm doing this every 24 sim hours and the non-infant milestones have way less impact on gameplay so it's kinda whatever.