Welcome to the forum!
I'll admit...your question and photos seem pretty clear to me.
That subwoofer/amp module from your LG home theater system could be used, but it's not straight forward (which is why you asked, I'm sure
). If you don't live somewhere that uses 200-240V outlets, then it's probably more trouble than it's worth unless you already have a transformer.
I don't know the simple answer for directly connecting your receiver to the sub module. I've used a system like that by taking it apart and making my own connections. The trick is knowing which pins on that input connector are for the subwoofer signal. Do you have the controller for that LG system anymore? If you do, and if it has any analog audio inputs, you could hook the receiver to that controller, and then the controller to the sub module, and use it that way.