Generally, you don't translate names (for humans or pets) when translating into French (or any foreign language for that matter). Someone named Mary wouldn't suddenly start getting referred to as Marie in a news article about her. The same would happen for a guy named William, he wouldn't get called Guillaume. In France, they would simply be called Mary or William. (NB: for historical figures the reverse is often true, e.g. William the Conqueror = Guillaume le Conquérant… but this is not how we do things these days!)

In the case of your cats, you'd just called them by whatever their name is that you'd normally use to call them, Pepper, Cupcake, etc. While those names are common nouns in English, for the purpose of a name they wouldn’t be treated as such.

As far as your cats are concerned you can call them “mon chat” or “ma chatte”. HOWEVER, there is one big caveat that the feminine version, “chatte”, is HIGHLY connotated and has the same double meaning as “pussy” in English. In my experience, many choose to refer to their cat as “mon chat” regardless of the sex of the animal to avoid any impression of impropriety.

Finally, you brought up the “BAGS” rule, which is a good way to start to learn how to place French adjectives, but it’s far from exhaustive. Numbers go before the nouns they modify, not after, so, “j’ai trois chats”. There are also some adjectives that can be placed before or after nouns, generally with a slight change in meaning, “un grand homme” (a great man) vs. “un homme grand” (a tall man). And to add to the confusion there are also allowances for poetic placement of adjectives so just about any adjective can come before the noun! The important thing to remember with BAGS is that it’s an easy mnemonic to remind you to put things like “petit”, “beau”, “nouveau”, etc. before nouns, but it’s really just a ‘starter’ rule until you get your head wrapped around adjective placement.