The custodian is providing bad information if they are saying that the timing of the 2023 contributions has any relevance to determining whether or not an excess contribution was made. An entire 9/12 of the 2023 annual limit could have been contributed all on April 15, 2024 and it would not be an excess contribution. The type of coverage on the date the contribution is made irrelevant. All that matters is that there was 9/12 of a year of eligibility.
Section 223 of the tax code is says:
(a) Deduction allowed
In the case of an individual who is an eligible individual for any month during the taxable year, there shall be allowed as a deduction for the taxable year an amount equal to the aggregate amount paid in cash during such taxable year by or on behalf of such individual to a health savings account of such individual.
(b ) Limitations
(1) In general
The amount allowable as a deduction under subsection (a) to an individual for the taxable year shall not exceed the sum of the monthly limitations for months during such taxable year that the individual is an eligible individual.
You can see from this that eligibility determines the amount than can be contributed, not the timing of the contribution.
