Your second to last paragraph I take to be your objective of all the foregoing writings. If I were to explain the details in the beginning of your post, it would take a lot of writing, perhaps too long in one response.

The two pairs of speakers have different sensitivities. They will play at different loudness with the same power input. So putting one pair on the left and the other on the right would be unbalanced. Turning the balance knob to compensate could work, but then you need to increase the volume to get back to the equal the loudness of the louder pair.

Some TVs have an audio output. Mine has stereo output in mini 3.5 mm jack, or optical fiber output, for example. Some have HDMI connections that output the audio. If you don't have HDMI, then this option is not helpful to you.

It seems to me you can get an late model AVR (audio-video receiver) new or used that can handle video and home theater surround sound. That would use both pairs of speakers and have an input for TV sound too. A powered subwoofer (or two) is highly recommended to relieve your speakers from trying to reproduce deep bass which requires lots of power and leave the rest of the channels to the AVR which is less demanding.

If you want to keep using the existing amplifier, one alternate option is to get another stereo amplifier, and a Blu-Ray player that outputs to both speaker pairs through the two stereo amplifiers, one pair front and one pair rear. The player will control the overall volume to all four channels, while the individual amplifier volume controls settings will balance out the front and rear volumes. This is more complicated to do, but it can work.

If your want to only use your current amplifier, and that amp has two speaker output sets - one for pair A and another for pair B, then you could use both pairs in this fashion, if there is some balance control for equalizing the loudness of A and B pairs of speakers. I would assume you have a selector switch for playing only A or only B or both.