Yes, it would sort it - but you don’t need to do that. You can just delete the untagged entries if they are duplicates of the ones you have tagged.
If you have bank tagging rules set it, it could potentially log the payment to the client as a “prepayment” and you could then pay down the invoices manually using the account credit. It depends on how you work really.
The feed runs each morning at around 3am.
This could also work. When are the payments made exactly? Although the feed is run each morning, you can manually refresh it if you think the transactions have cleared (view the bank statement, More Options >> Refresh Bank Feed).
We added an option to send an email from the bank tagging process a while back. I’ll find the topic for you shortly and update my post 
[Edit]
As promised:
Email receipt of payment from bank tag - #6 by Glenn