AV Receiver with Integrated Amp - Sub Connections?

If I want to have the sub work with my receiver while watching movies and also the integrated amp while listening to music, how do I connect the sub to both the AV receiver and integrated amp? 

Also, will this not work since an integrated amp doesn’t have a crossover setting for a sub?
If your sub has speaker/high-level connections, I guess by definition it’ll work for both in this setup since everything goes through the integrated amp to the speakers. Personally I’d optimize it for 2-channel and just let those settings fly for HT whatever that gives me. The downside is you won’t be getting the LFE signal from the AVR so you’ll be missing out on some of the HT experience. For me this would be a fine compromise since stereo is more important to me and I think HT performance would still be very acceptable, but YMMV.

If if your sub doesn’t have speaker-level connections, this goes beyond my already meager understanding of the subject and I got nuthin’.

In a perfect world I guess you’d have a sub with both speaker and line level inputs and use one for stereo and the other for HT and be able to switch between the two, but I have no idea if such an animal even exists. I’m sure others more knowledgeable on this subject will be much more helpful and correct any errors in my recommendations, but just wanted to get something started here.
Thanks Soix,

I have been thinking about it and it seems like what I am trying to do can easily be solved by a product of which few exist....An integrated amp with HT bypass AND bass management.  I did find one so far called the NAD M2, but it’s too rich for my blood. I could have plugged it into my receiver and had the best of both world and connected my powered sub to the receiver and NAD M2 and switched between the two with the press of the bypass button. How did I, somebody who literally entered the audiophile world a week ago, think of something only to realize the manufacturers have not capitalized on this yet and very few exist!?? Anyway, I assume we will start to see the concept takeoff. I am not sure, but the PeachTree Decco May also provide this solution in a box to crossover between the AV world and music world. For a while it seems like people who want these features will be stuck with the sound of the few companies that are making them. In the meantime, I may need to look at a pre-amp with bass management to go between the receiver and amp. 
There are a bunch of Parasound HALO Integrateds up for sale.  Several are under $2k.  The subwoofer crossover is not controllable from a remote.  However, it can quickly be switched on/off on the back panel.
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.

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.