AV Receiver with Integrated Amp - Sub Connections?

If have room, I would just get another sub if you care about the LFE & ULF of your HT.  I tried doing what you were doing but somewhere along the chain, I couldn't make the sub work for both configs based on the bass I wanted.  If you don't connect the sub to the HT receiver and use Audessy, the sub won't rumble like you expect since Audessy just sees your L/R as large speakers.  You'll get bass, but I found it to not be as impactful as the sub connect to the HT receiver.

I opted for the Monolith 12.  I dedicated my HSU to my 2ch and glad I went with the Monolith for my HT.

Here’s the best solution Integrated Amp available on the market (link below)...Might make you want abandon the surround sound receiver, center and rears you have and convert movie watching to stereo with a sub.  
Or, you can upgrade from your receiver/integrated combo to a better overall receiver or go to a separate pre/pro and power amps.
https://www.anthemav.com/products-current/model=str-integrated-amplifier/page=overview 

And DOES HAVE Home Theater Bypass FYI...

I think I get what you want to do because I’m planning at some point to do something similar. It blows my mind that a lot of companies have “home theater” pass throughs but never do they have sub inputs and outputs which is nuts as subs are critical to home theater. Maybe they assume you won’t be using subs for two channel which I think is also silly.  Take a look at the JL Audio cr-1 crossover and see if that does what you need. 

My NHT SubOne i has an external controller that has inputs for both LFE and Audio. You can connect the receiver via the LFE/subwoofer output and you can connect an integrated amp, or pre-amp, via an analog line-level output. You can set up each connection independently. I don't know if there are any other subs that have this option. Perhaps there are some subwoofer controller that offer this type of connection.