I maybe wrong but I though the reasons the new switch had better battery performance was due to the newer chip-set rather that an increase in battery size?
I might be wrong but I think you can transfer saves by moving them to a USB but you will need the profile to be on the PS4 to actually use them and I don't think the profile is transferable?
Im pretty sure it wont let you restore the back up unless you upgrade first. I think it will bring up a message warrning of this before restoring but i wouldn't risk it personally.