Splitting a mnemonic code or seed like that is not a good idea at all.
Using a secret sharing scheme is superior since it does not leak any information about the secret at all.
Using a secret sharing scheme is superior since it does not leak any information about the secret at all.
Right, but the secret sharing scheme has two main issues: 1) amnesia (you forgot where did you store one of the 3-4-5 pieces) and 2) what if one of the pieces is somehow damaged.
Regarding 1), you can't protect against everything.
If you forgot the locations where you stored the mnemonic code, you might as well forget how the service (your service) is called. Or that exchanges exist. Or you might even forgot that bitcoin exists.
That's a bad argument IMO.
For 2), that's what a N-out-of-M scheme is for. You might have a 4-out-of-7 scheme. Or a 2-out-of-5 or whatever.
Exactly! This is why you most probably need to destroy the private keys of wallet A right after you signed the transactions. 
Another problem with your approach is, that the user will depend on your 3rd party service.
Given that a user follows what you describe, after not having any possibility to access his coins, he relies on only your 3rd party service to broadcast that signed transaction.
The whole setup allows you to blackmail the user following your advices.
While the user could save the signed transactions himself, the user you are targeting with such a service probably doesn't even know what a signed transaction is.
Even without blackmailing, he relies on you not losing the signed tx.
Moving from proper risk management to relying on a 3rd party is... not how it is supposed to work.