I can't find any examples online (and it seems unlikely), but is it possible to attach two Burley-type trailers together for the purpose of towing four young kids (ideally behind a tandem)? None of our kids are old enough for the tag-a-long type bike for at least another season or two, so they all need to be in trailers.
While we did it temporary with two cheap (InStep) trailers and I would not do this for any longer period of time. There was no really good attachment point to secure the second trailer to the first one, I ended up using several bungee cords and the 1st trailer frame flex was noticeable. Pulling the train was OK, comparable to pulling TAB+trailer, slightly easier actually, as I did not need to keep TAB balanced upright.
Since you mentioned tandem, my advice would be to stick with two single bikes and two Burley-type trailers. How big are your kids?
Also, if money is of no objection, be aware that there are several European multi-kid trailer manufacturers, some of them seating up to 10 kids.
I've seen it a few times but I have never paid attention to how the trailer to trailer connection was accomplished.
The tandems two trailers could work for your season or two but after that you'll run into problems with seating positions. I.e., with one TAB you still need two trailers unless you can get a 3+ seat trailer. If you have 2 TABs, you will also need a trailer. Your more flexible options will involve a second tow bike.
You can use a seat with a burley trailer, that worked on ours, the rack mount for the seat wasn't in the way of the trailer hitch. I bought one of the trail a bike seat backs and our daughter did fine at 3 with it.