You've already made your (correct) decision, but to add to it, I got tested a little over a year ago. I carry a lot of my weight in my face, and I need to lose some weight, so it is almost certainly fat related for me. I was snoring whether I was on my side or back, it didn't matter. When I went in for the test they hooked up a bunch of electrodes all over my head (made easier by the fact that I shave my head) and then had me go to sleep without anything hooked up except those. After it was obvious to them that I was, indeed, snoring, they hooked me up with a CPAP and then did additional monitoring throughout the night. No nostril tube or anything like that. (It should be mentioned that a couple of my coworkers that have Kaiser only had to do the wrist band thing at home.)
I use the "nasal pillows" and fall asleep on my side 100% of the time, and it isn't a big deal for me. I got used to it within a couple of days, although they didn't supply me with the additional humidifier at first and it dried me the fuck out and I'd have a runny nose all day from it, so specifically ask for a humidifier. Even if you end up not using it, better to have it right from the start instead of having a bad experience at first.
One of the Kaiser coworkers says he tries to use it but ends up tearing it off in the middle of the night, and not even using it 5 days out of the week. To me he might as well have just flushed his money down the toilet. If you get prescribed one, use the fucking thing, it really does work. Granted, I have terrible sleep habits in terms of staying up too late, but that's a separate issue. Hell, I use the damn thing even if I take a short nap, because it doesn't bother me and it stops the snoring. Make the effort to get used to it and then stick with it, you'll be happy you did.