Bi amping with two different pairs of monoblocks

Every case is different.  I don’t agree that it’s a ridiculous proposition.  It depends on several variables. 

In my case, the Dynaco/VTA monoblocks benefitted from seeing an 8 ohm load vs 4 ohm, and also benefitted from not having to operate in the lower octaves.  Bass improved from the solid state amp too.  YMMV, but you’ll never know til you try. Good luck!

Another option would be the Dayton Audio DSP-408 or something similar. It gives you 4 channels in and 8 out and a multitude of controls, including volume and crossover . It's $179. It's overkill, but will work. Besides, you can play to your heart's content. 

As mentioned level matching will be the issue you need to address between the amps.  An external crossover is the best way to address this. 

Instead of this method of bi-amping that you are considering, how about a different "bi-amping" approach that might yield as good if not better results?

You don’t mention in your post if you have any subwoofers but here is an alternative approach, given that your Triode TRX M845 tube monos have a low damping factor relative solid state and thus controlling bass drivers is not their forte (but musicality is is - in particular - the midrange):

High-pass the signal out of your preamp at, say 80 Hz, and let your tube amps and main speakers handle everything from 80 Hz and up.

Tap the lineout from your preamp and send it to two very good subwoofers that take care of the duties from 80 Hz and down.

This way you’re using the DSP in high-quality subwoofers paired with their solid-state amplifiers, and designated woofers for the bass, which should yield you a better and tighter bass response. You also have more freedom in where you place your subs to get a more even frequency response throughout your listening room.

So, instead of spending money on the second set of monoblocks for your mid and high frequencies, spend the money on high-quality subwoofers instead.

Yes, your main speakers are very good but they can’t touch the bass reach and response of high-quality subwoofers, not least since the subwoofers will be powered by solid-state amps with a significantly better ability to control the drivers than what tube amps can.

@knotscott

 Every case is different.  I don’t agree that it’s a ridiculous proposition.  It depends on several variables. 

I said it was a ridiculous proposition in the context that the OP would be better served buying a decent front end and some high quality speaker jumpers, they really make a difference.