Finally resolved. Quick update for anyone else with the same problem in future years:
After a few emails back and forth, they told me to submit under the Department code 1903 (Public Policy Analysis). This is, again, contrary to whatever is listed on their website.
A month after that, with no update to GRE status on the portal, I re-sent my GRE registration number to admissions. That seems to have finally solved it.
@aaprabhakar Thank you for the advice!