>>26713203>Would the time cycle just go on as if nothing had happenedThis is what would happen, but the clock cycle may be out of sync with reality. It may or may not prompt you to reset the clock, and if it does, it'll only do it once.
Also you may need to fix the berry glitch to get berry growth working properly, but practically everything in gen 3 past RS has the patch to this, so just trade with Colossseum/GoD, apply the patch from FRLG/Emerald, or scan an e-card from an e-reader/Pokemon Channel/Pokemon Box RS.
>>26713235No Gen 3 game requires the internal battery to save. You will never lose your save from replacing the battery in RSE. If it says it will then your cart is a fake.