Bart is one of those names where the cultural association is so total that almost no one reaches for it without the reference in mind. Bart Simpson has been on television since 1989, and in that time has thoroughly colonized the name's cultural space. At rank 1295 in the pet registry, every Bart in the dog licensing database almost certainly arrived via Springfield, Ohio.
The Simpsons Vector
Bart Simpson — perpetually 10, perpetually in trouble, perpetually loyal to his friends despite his best efforts — is one of the most defined animated characters in American television history. The personality projection is clear: a Bart is energetic, not entirely obedient, funny, and fundamentally good-hearted under the mischief. Beagles and Jack Russell terriers are the breed equivalents. The human name's history predates the show at /names/bart.
The Name Before the Show
Bart derives from Bartholomew, an Apostle's name with Aramaic roots. The pre-Simpsons Bart was a dignified, slightly old-fashioned shortform — similar in register to Bern or Gus. The show reset all of that entirely after 1989. You cannot now name a pet Bart without the cartoon, and there's no real reason to try.
The Counter-Reading
Bart is a strong choice for an energetic, mischievous dog — but owners who want a name they can say at the vet without prompting a knowing smile should look elsewhere. The Simpsons reference is permanent and inescapable. That's either a feature or a drawback depending on your relationship to the show. Compare Homer for a deeper cut from the same universe.