Export loads images needed per animation. Using smaller images is not necessary.
Animate mode should not affect export, but it seems there may be a bug. It's great you found that the problem goes away, but we'd like to be able to reproduce the bug so we can fix it.
I put physics on spineboy's head, so it's very floppy during the jump animation. I exported a PNG sequence for jump from animate mode and setup mode with Deterministic
on and the PNGs are identical. Are those the steps you are taking? I wonder what it is about your project that is different?