Can I immediately file an amendment to a tax return when payment is owed on both original and amendment?

1. can the financial outfit correct the deposits to be for 2023?

2. if not, yes you can amend 2023 taxes and have the payment be made for the difference as long as the first return has been accepted

If the IRA administrator made an error then I would ask them to correct it and apply the IRA contributions to the correct year. My experience is that they usually will correct it as they want to keep their client happy.

I don't think the IRA trustee reports to IRS which year the contribution was designated for.  Your client can tell them before the 2024 contribution is made, I'm sure they will take the money.  

It seems the payments were made timely to the tax year 2023 contribution deadline, so that would be designated for which tax year and this is reported on Form 5498. If the taxpayer made the error, it's really easy to recharacterize it. I've seen where the IRA provider offers this function on the management website, even. If the provider did this, they also can recharacterize it.