Here’s the key point re: DCTAG for your situation—you must “be a District of Columbia resident for at least 12 consecutive months prior to the applicant’s first time in college and maintain continued {domicile} throughout the applicant’s college matriculation (Note: Dependent students’ [under age 24] domicile is established through the parent or legal guardian).”
Where does your child go to school? Public or private? If they live with you and your ex 50/50 and attend public school, I would assume the public school’s location would determine their residency for DCTAG purposes. If private, it’s harder to say, but the school should be able to advise.
DCTAG has existed for 25 years, always at the 10K/year level; the point was to try to make up for DC students’ lack of in-state options for college, but of course that $10K per year now does not go as far as it used to. It is funded by Congress so subject to the whims of Republicans, who have threatened to eliminate it many times over the years. Given the current administration’s posture toward DC, I would not count on its long-term survival. There’s also a household income limit of $600K/year for eligibility; if Congress doesn’t want to kill DCTAG outright but wants to make it less accessible, significantly reducing the income threshold would be the most likely move.
I don’t think there’s any application advantage to one jurisdiction over the other; there are tons of highly prepared students applying from both, and the main factor would be your child in their school context.