The advantage to an outboard amplifier like that behringer is that it saves on woodwork, so there is less cutting involved and that makes for a simpler construction. It is also quite a bit less expensive than a comparable plate amplifier, and what's more, it is easier to keep cool. The only disadvantage is that now you have to put it somewhere since it is not attached to the subwoofer cabinet.
You would not want to put that amp on top of an AVR, in fact you do not want to put anything on top of an AVR. That AVR needs ventilation, and blocking the AVR vents with a warm amplifier is the last thing you want to do. You can just place that amp on another shelf and it will be fine. Just make sure it has some ventilation room.
If those 15"s are too big for you, here is an idea, get four of
these Tymphany 10"s (on sale now for 50 a pop, but that sale ends tomorrow). you can string 2 of those per channel on the Berhinger amp since they are 8 ohm drivers. You can make two boxes that each have two drivers in them. That amp can run those 4 tens with no problem.