Think about this: You have a compound sentence and the second clause is introduced by a prepositional phrase. How do you punctuate it? How about this?
The dog ran after the cat; and if the cat didn’t run fast enough, the cat would be caught. This is a little old school, and I may have shared it before, but many of the writers in the late 1800’s used the semi-colon like this. If removes the clutter of too many commas. We seem to comma ourselves to death many times.
Another tip. The rules let us forgive the use of commas in short introductory words and phrases. Why don’t you save yourself the time and trouble of figuring out what is too short or too long to be offset by a comma and just do them all with commas. Good luck!