If I'm playing a movie through my AVR, won't the AVR need to be connected to the integrated so that the movie's audio makes it the L/R speakers?
Yes. But that's exactly what I told you NOT to do.
Movies, whether from CD, or Blu-Ray, or laptop, whatever, L/R goes straight to the integrated. Obviously that would be via RCA. Your movies probably go HDMI from whatever into the AVR. Whatever. That's one reason HT sounds so crappy. HDMI. Twenty hair thin wires, video and audio, all crammed in tight together. Sheesh. No wonder it sounds like crap. Convenience over performance.
Anyway its HDMI from your movie source to your AVR. Connect only surrounds (and center if they talked you into that one, sorry) to the AVR. Do not connect stereo speakers to AVR. Then L/R audio from your movie source goes straight into your integrated. Connect L/R speakers to integrated.
Now use the integrated for main volume. Set your main volume where you want it. Then use the AVR to set surround volume. You will NOT need a center channel, that's what stereo is for, so if you have it connect it or don't, whichever you like.
Use the AVR to adjust surround volume to what you like when the mains are at a good volume level. If you turn the mains up louder the surround will be a little less. I doubt you will notice unless you are really trying. The minute you stop trying and actually watch the movie you'll forget all about it. If you turn the volume down a lot the surrounds will be a little loud and annoying. This is actually a good thing. The combination will gradually over time make you aware of just how gimmicky they are.