It is not really extra effort. He stops paying until he is informed of her current address, at which point he serves her the papers. If he never receives her address, he never makes another payment and no further court action occurs.
There are a few problems with that approach. First, there are already means in place to serve notice when a current address is not known; this erely move stat into the digital age. While there are problems with this approach it's no worse than requiring a notice in newspaper at the last known address; i'd argue in some ways it's better since there is at least some reason to believe the account is active and tis may see the notice. Secondly, suspending payments would not necessarily end his obligation to pay and thus he needs a court order ending the liability for payments. Absent that a court could order him to make the back payments.
Of course, with the ruling I suggested in hand, he might also have a basis for requesting a court order to freeze the account in question, since the address on the account is not that of the registered account holder.
Even so, the person has not been served and thus might be able to fight any legal cation on that basis, leaving him still on the hook for payments up to a final decision. As for freezing the account because oaf a bad address banks have plenty of accounts with bad addresses and all that happens is eventually the state gets the money.
Even if he cannot obtain a court order, he can possibly make the bank uncomfortable enough about the account that they report it to the appropriate government agency for investigation as a "suspicious account" under various anti-money laundering laws.
I doubt a bank, let alone a law enforcement agency, would get worked up over a few hundred dollars being sent to an account with a bad address. Even so, a call to the person making the deposits would reveal they are court order child support an thus case closed.