Has the new kitten been tested for FeLV AND FIV yet? The Vet should have suggested this to you and explained why its very important when bringing in a new cat of unknown origin into your household and potentially exposing your presumably healthy established cat. FeLV especially is very contagious and kittens,can acquire it from the mother.
If the Vet didn’t say anything about this, I would strongly suggest looking for a different vet. There is a “snap test” which can give results in around 15 mins. or so and is usually done right in office as part of a typical first visit.
You should really keep them separated completely until you know the kittens status for the sake of your established cat.
I ALWAYS get this done prior to introducing a new cat. I normally adopt shelter cats and many times they have already done this since it is so contagious. But several years ago, I adopted a cat from Craigslist. Even tho she was already two years old and they claimed they had kept her as an indoor cat, digging a little further revealed that she had escaped a few times and gotten pregnant. When the Vet did the test, unfortunately it came up positive and I had to find a new home for her since I already had another 18 yr. old cat who was healthy, even tho aging and there was no practical way I could continue keeping them apart indefinitely.
I was really glad I had that test done since they had had absolutely no contact for the initial few weeks I had the new cat prior to the Vet appt. The new cat didn’t look sick or anything and theres really no way to tell other than the test.
Hopefully you’ve already had the new kitten tested. But if not, I would strongly urge you to do this ASAP for the sake of your established cat.