Just an update in case anyone has this issue in the future, my top is functioning 99% normal now. I should've written down every detail of my efforts with the top, but I can boil it down to this.
With the previous owners leaving the battery dead for multiple months, my hunch is that the top module lost track of its position.
Over the course of a few days once I knew how to use the connector to get the top to retract, I'd release the cowl motor using the Allen key and connect the wires to get the top to go down. My gripe was that the top didn't latch in the down state.
At some point, I tried connecting the module when the top was mostly down and using the interior down button. To my surprise, the car took over and completed the retraction process, locking it in place and turning off the flashing red LED. Also, the top would go back up without any problem.
I used my wire method a number of times and finally noticed it would only need to retract ~20%, and upon plugging the module back in, it'd finish the process. Finally, after using the interior buttons to put the top up one time, I used the top down button and it retracted without any real drama.
Either all of this relearned the position of the top, or it needed to be in a different starting state for the module to sort things out.
So anyone who has this, try connecting those pins and leaving the top in different positions. Each time, plug the module back in and see if it starts to work. There may be a sweet spot where it all clicks and the module is back to normal after a battery replacement.
The only quirks that exist now are probably specific to my car; I'm guessing it needs a hydraulic fluid top up. When the top starts to go down, it needs a little pull towards the rear to get it moving, like the motor can't overcome the initial force. And when the top is finishing the down process, I need to apply pressure over the rear latch area for it to catch. Maybe some air in the system make it 'spongey' and it can't exert the force required for these couple steps.