How can someone recoup leave if their DA 31 for non-chargeable leave (HRAP) was signed by an approving officer of insufficient rank?

Which O3 signed? It could be completely on the up-and-up or a mistake on the part of the officer. If the O3 was the designated representative of the O5 commander (for instance, when I was a Battalion Commander, the S1 (a CPT) had designated authority to sign for me regarding leaves and passes) there there shouldn't be an issue. If this was the case, then all that is needed is to submit documentation that the O3 was able to sign for the O5 in this case. However, in most cases the document they are signing would have my signature block and rank typed out and they would do the normal "for" in front of it and sign.

If it wasn't the situation, then as others have stated, it's time to go up the chain and get this fixed. If he has a better relation though the NCO chain, then take it that way to advocate on the Soldier's behalf (I know many lower enlisted would feel very uncomfortable walking in to see the 'old man') if needed. I would advise that he talk to his chain and whatever leadership is supposed to be working on the issue and let them know he's doing this. This will either 'prod them' to resolve it (if they've been putting it on the back-burner) or at least let them know (who knows - they could have been falling into the trap of "fix it yourself and never ask for help, even when you hit a roadblock you can't overcome").

Regarding how it is corrected, that's simplicity. As long as you can get the O5 on-board (or his designated representative) that the Soldier shouldn't have been charged for the leave, then another DA31 is submitted to cover the period and the situation is explained in the remarks.

Who in the chain of the command has he brought this up to? It clearly needs to be escalated to the next person. This isn't the first or last time something like this happens. It shouldn't be hard to grab a signature and have the information fixed. Open door policy is there for this kind of thing. I'm also surprised this wasn't brought up to the recruiter that the recruiter assistance was helping. They generally can reach out to expedite this stuff too. Like I said, it's a common issue and shouldn't be an issue to fix but you just have to keep escalating it to the next leader if it isn't getting addressed.

This individual has to lean into this. This is their issue to fix and the responsibility to run it up the flagpole is theirs. Don't over extend yourself on this because your ability to address it is limited.

Why doesn't he just ask for an Open Door with the first O5 in his COC and have them sign it, then resubmit it? Has he spoke to his CO? First Sgt? CSM?
Finance is correct that the DA31 isn't valid, but all the soldier should need to do is have it signed off on by an 05.
Edit: O-5 or their designated representative. If the original O-3 was designated, all that would be needed is a memorandum that stated that fact.