Yeah, that last little bit is tough. There will always be a little left since, as you mentioned, the bilge pump can't get all of it.
Can you reach the garboard drain plug while the boat is on your lift? Maybe add a plank so you can get to it? I've seen many lifts with "catwalks" so owners can get around the boat for maintenance while on the lift.
If not, the shop vac is a good idea. You could also just use one of those super absorbent cloths - lay it down there, wring it out, lay it down there - probably would only take a few times. If you had it on a stick/pole, you wouldn't even need to reach down there.
Or, plug the holes in the cooler and as the ice melts, thoughout the day, just use a cup to scoop the water out a few times over the day. At the end of the day, the coolers will be easier to dry than your bilge.