Does an Invalid Transaction persist in the mempool?
A Wallet tries to spend more BTC than it current has


Technically, that would only be possible, if you try to spend an input (coin) that you can't sign (a.k.a. that you don't own).


Your local mempool (if you run Bitcoin Core or a similar fullnode) would reject the transaction immediately.
The transaction would not be broadcasted over the p2p network, because all nodes do reject transactions that are trying to spend inputs with invalid signatures (or inputs that doesn't exists)



  At some much later point, the Wallet gets more value.


The transaction is already made. It won't auto-update. It was already rejected.

Now one expect your local wallet knows the ("old", invalid transaction).

The only thing you can do now is to recreate the transaction (now with the correct inputs) and have it send over the network (and have it mined).


  The previously rejected transaction can suddenly be confirmed.


Invalid transactions cannot be mined and will not be broadcasted (broadcasting would be possible but extremely unlikely).