The AP stylebook (which I have a copy of), as far as I can determine, does not address the specific construction you describe. The AP stylebook mainly addresses when to capitalize the names of political parties.
As I said, the GQ answer was given in the second post. You appeared to be more interested in debating the use of the term rather than accepting the answer. Hence I closed it.
Since the use of the term has more to do with politics than grammar, if you want to debate how the term is used, open another thread in GD.
Apparently NYT disagrees: “As the Senate moved closer to a final vote on a Democratic-sponsored bill, lawmakers spent the day trading stories about how they struggled to raise money for their own campaigns and how the race for contributions should be halted.”
I would argue that “Democrat-sponsored” is the preferable construction, but English does allow for noun+past participle and adjective+past participle constructions. I think, in this case, the “-sponsored” should follow a noun, so “Democrat-sponsored.” Searching Google news using “Democrat-sponsored bill” and “Democratic-sponsored bill” in quotes shows the former being the preferred construction in recent Google news sources, at 228 results vs 189 (with the latter even triggering “Did you mean Democrat-sponsored bill?” from Google.)
As I said, as far as I can tell the AP stylebook does not address the issue.
ATMB is not the appropriate place to debate the issue. Since you appear to want to debate the issue, if you wish to do so, as I said you can open a new thread in GD. I’m closing this one.