The only progress I've made so far is this:
Based on the suggestion of a TAC Engineer, run the following command to enable class 4 power negation via LLDP -
#configure lldp ports [port] advertise vendor-specific dot3 power-via-mdi with-classification
#enable lldp ports [port]
At this point the AP registered to our controller still reports PoE Status as Medium Power (15.4W) under the advanced tab of the specific AP.
Disabling CDP under the AP settings and rebooting the AP will then display a PoE status of Full Power. The switch at this point will show higher wattage than it had before (I've had up to 8.4W today, vs mid-6W range earlier), but it's still not what I was expecting to see. Cisco's Prime Infrastructure monitoring still reports that the AP is drawing low power.
It still doesn't feel like the AP is getting the power it's requesting, but maybe a step in the right direction.