I did resolve the problem. It was indeed the body control module as suspected. I can't speak for the 2011 model Avalon, but I can tell you my experience. Unfortunately I did not think to take photos while doing the swap but I can describe the process. Finding a BCM either at a salvage yard, off an auto parts store, or through the dealer would have been several hundred dollars. I instead got under the dash, managed to take a photo of the label on the back of the BCM which is bolted into the back of the junction fuse box. I then looked on eBay for the junction box with a BCM number matching that of mine. There are different BCM model numbers, which I assume pertain to the different features they are programmed with, such as sun roof, no sun roof, etc. I am not sure if they are interchangeable, and didn't care to find out. The entire junction box with embedded BCM was only $38 shipped, and I have a spare junction box with extra relays and connectors to boot.
You'll have to undo four screws to remove the plate covering the module. Then you'll need to gently lift the module up with a tool, being careful not to damage the chips and copper leads on the underside. The module is held in by tension with a long and narrow connector linked to the junction box. Try lifting and rocking it side to side. It will flex a little but don't overdue it, and keep the lift pressure on both sides to evenly separate the connector. Eventually it will pop out with enough equal force. I didn't need to remove all of the cables for my junction box in my car, rather I undid the three bolts securing it to the frame and undid a few connectors on the back to give it enough slack to pull down to remove the module and swap. You will need to remove the plastic panel and metal bar on the bottom which holds the ODB port as it blocks the box from the front. It's as simple as popping in the new module. For newer vehicles this may require reprogramming of the remotes if the transmitter is built into the module.
End result is all my locks work properly for about $38 and about an hour of time. It will be half that time if you already know where all the bolts and plastic tabs are that you'll need to remove.