This is a very unpleasant situation indeed. As said by Sehgalanuj, I am sure the key thing here is that you were flying VN and that VN is using another lounge in BKK, so AF cannot bill VN for your kids lounge stay. It would have been likely different if VN was using AF lounge in BKK. Now that I have said that, the policy is definitely unclear and of course I think this attitude from staff is ridiculous toward top tier customers and you should definitely complain. For quite some time now, it’s not anymore AF staff at the lounge but subcontractors and they are a mix bag (some are great, some are average to poorly trained). It should at least trigger a clarification of rules to teams or for what is written on the websites.

just to share one bad and one good experience in this lounge :
Something like 8-9 years ago, I was flying to BKK in J, connecting on PG in Y (no J on ATR) with long connection (like 3 hrs). I had purchased the P ground services in CDG and while I was at the P lounge, I discussed with the lounge manager about my trip and asked him that it would be great for me to be allowed exceptionally in the BKK AF lounge for my connection, rather than being stuck in the not-so-nice PG Y lounge. I was already PFL at that time, flying J to BKK with paid P ground services, I thought that would be good justification for this gesture. The P lounge contacted BKK and the station manager replied positively and that the lounge staff would be informed. Guess what…I showed up at the lounge and they looked at me as if I was coming from Saturn or Jupiter and they denied me access and refused to call the station manager to check. I was really furious .

More recently (February), flying AF BKK in J and also connecting to PG but on separate tickets (I know well the risks but I had no choice than having separate tickets). Due to the delay in AF flight, I missed the PG flight and had to change my ticket and had to now wait 7hrs in BKK . Being Ultimate, I decided to stop at AF check-in/ticketing counters to ask whether I could go to AF lounge to wait, explaining them the circumstances. I was not optimistic at all knowing previous experiences here but they made a call to whatever supervisor and it was accepted. And the lounge staff was very nice with me during all the time I stayed in the lounge.