This can be done - I have done it, but it depends on the hook up options on your equipment.
1)your sub has separate inputs for LFE *and* L/R Line Level inputs (not a combined LFE on Left input option)
2)your integrated amp with HT Bypass has line level pre-outs.
If so, all you need are some interconnects to go from your AVR Sub out to the LFE input on the sub and a pair of ICs to go from the pre-outs on your int amp to the L/R line level inputs on the sub.
For Home Theater, the integrated amp gets switched to "HT Bypass" and the pre-outs should be disabled as the preamp section of the int amp are bypassed and it acts as a power amp only for your Front L/R speakers.
The sub LFE input will get the bass managed signal from the sub out on the AVR.
Enjoy the movie.
For 2-channel music listening, turn off the AVR - there will be no sub out signal going to the sub.
Select whatever source you want to listen to you on your int amp and the preouts will feed the sub the L/R signals. Of course, you will have to manually integrate the sub xover with your mains and your mains won't be high pass limited and will get the full range signal from the int amp - but I have found that with time and effort you can properly integrate mains and sub this way esp for music, since that won't have any scary and dangerous LFE signals ;-)
Enjoy the music.
No switches, no cable swapping - slick as could be.