I've long used those same basic bindings. I found that aggressively telling MacOS itself to not do ANYTHING with three fingers helps a lot on the first issue. I can't remember what the defaults are - I've clearly changed them - but when I change computers, I know I have to visit all three tabs of trackpad settings to get MacOS to NOT claim triple tap so that BTT can have it.
I fought the four finger issue for a long time and don't know if it's still applicable, but since I developed the muscle memory for it 8-9 years ago, I've just stuck with my solution. I just use four finger double tap to bind to CMD-W. It's still intuitive enough. You can convince yourself that since it's a semi-destructive action, it SHOULD take more effort than opening a tab. You'll find SHIFT+CMD+T a useful shortcut to remember, too...
The idea is to let EITHER the browser/OS handle triple tap (booo) OR let BTT (yay!) handle it. If both recognize and act on those actions - especially if they have sligthly different timing - you get exactly the symptoms you're describing.
I change machines a lot less often than I used to, but I generally consider this a pretty reliable set of bindings for me. I remember seeing the kind of wonky behaviour you're describing, but I've used BTT for so many years that I can't remember what combinations of versions and OSes were problematic. Ventura on my 2020MBP and and external BT pad on Catalina (?) on a 2012 iMac are fine. I'd assume any combination between those is fine.
BTT stops working after sleep or close lid still nails me at least once or more a day, but that's a different issue.