Kudos to see where this goes. Few things.
Since you have them in carboys with minimal (zero?) headspace, I'd let them sit and get dusty. If you have any brett in there (and I would guess you might), it can take a long time (6 months +). You want the carboys to burp themselves over that time, so bottles would likely become overcarbed or even explode. I don't think racking underneath this has any chance of removing the infection, so you are in it for the long haul.
If you can swing it, I agree with transferring out of the AJ containers into a 1 gallon carboy for this long term fermentation/conditioning period. A little O2 is okay, but the AJ containers would probably give too much over 6+ months. Too bad you aren't close I have 3 extra 1 gallon carboys from my fall cider that are sitting around.
And lastly, to ensure you don't get future infections, I'd fill your bucket with cool water and add 1 Tablespoon of bleach per gallon and soak all of the equipment (including spigot) that touched this beer for 25-30 minutes. Then a few hot rinses and spray down with starsan. Should kill everything.