Follow these directions to link your Switch to your PS account (or vice versa):
https://forums.theshow.com/topic/49818/switch-successfully-linked?page=1
What your saying about downloading files doesn’t make sense, You need a copy of MLBTS for each device. After you’ve installed the game on each device, you choose which account (your Nintendo or your PS) will be your main account and link the other account to it. The accounts don’t merge. Both consoles talk to the PS servers and share the progression data. If you unlink your accounts, your main account keeps the progression and your sub account reverts to it’s previous progression state.