No, you don't have to amend your tax return. Because this is a SEP-IRA contribution, you as the employer, are the one responsible for reporting the year, to which your contribution applies.
I am not sure why your custodian would tell you that they don't report the contributions to the IRS, because they are required by law to do so by filing a Form 5498 each year. They do not report the tax year to which your contribution applies.
You can designate your contributions made between January 1 and October 16, 2023 (the extended due date of your tax return) to 2022. @jayd65
[Edited 04/05/22 | 4:32 PST}
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