Matching the width and thickness of the original trim might be a challenge, since mills are making trim so much thinner now than they used to. But yes, you can clean up the ragged edge of the casing and cut a new piece to fit. If you can't find a match, and the mismatch bothers you, you may have to replace the casing all the way around the door. Hopefully, the dogs will adjust to their new home soon.
Congrats on the house and
to the forum.