If I plant them now, will they grow after I change the battery?
The RTC and the save data timestamp would be different, causing the berries not to grow because the RTC would be at 1/1/2000.
Should I leave the berries? Or would picking and holding onto them for later be the best option?
You can pick the berries if you want, that won't affect anything, you just can't grow more for a while after you pick them.
Berries will not grow on a game with a restored battery unless you let the game sit for as long as your save data has been active. You said this number is 5 days, so that means your save data's timestamp is currently 1/6/2000, and your RTC once you replace it will be 1/1/2000. (Nor your save data or your RTC know what year it is, and it didn't know it back in 2002 when these games came out either.) Your RTC currently is void because your battery is dead.
The only way for you to grow berries in this scenario is to replace the battery and then wait 5 days until your RTC catches up with your save data and updates it, allowing berries to be grown again.
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