No amount of external pressure to the arm will cause pain. I'm thinking I might have some muscle or other soft tissue tearing or strain, though the pain feels quite close to the bone.
Any advice would be helpful. I'll likely just skip out on training sessions for a few days.
The that you have been experiencing in your arm might indicate that your muscles are in trouble or that some of your nerves are not really good. Usually there shouldn’t be any pain in arms when you inhale, no matter if you workout or not. I think that you might be having some problems in your shoulders. Have you considered talking to your doctor about this issue? How long has this been going on? I think that you should try to stretch your arm before and after your training and if this continues to happen you need to talk to your doctor. I hope that this was helpful in any way.
All the best,