Review the original item:
My cat Buster is always tipping over the garbage can
then he searches through the trash for milk jug caps, which he bats
around the floor, the feline version of playing ice hockey.
You wanted to fix it this way:
My cat Buster is always tipping over the garbage can,
but then he searches through the trash for milk jug caps, which he
bats around the floor, the feline version of playing ice hockey.
A perfectly legal move in writing is to connect two main clauses with a comma and a coordinating conjunction like but. The important thing to remember, though, is that you must choose a coordinating conjunction that logically connects the two clauses.
In the example above, but, a transition that shows contrast, is illogical.
You might want to consult the rules for fixing comma splices and fused sentences.