"The new custodian (my custodian) said they refuse to code it as a removal of excess contribution because the contribution did not happen there"

It's not contribution, so it's also not excess. Contribution is the New Money each year for a qualified amount.

You have an ineligible rollover. You need to "undo" by corrective distribution. The taking is the same, but the codes used on the tax form will be different.

The party paying out only knows it went out. Their code isn't always applicable to the reality.

If this all happened in 2022, there still is time to correct for it.

The first entity 1099-R is the first distribution, so that is Taxable. The redeposit is moot, and the second 1099-R is the corrective distribution; these two offset each other for the tax year.

Read about Corrective Distributions.