The employer is correct that if they were not the ones who made the mistake and you have not made an excess contribution, they are not permitted to reclassify the contribution.
Unfortunately, your proposed plan will not work because the excess would be deemed to be in the designated Roth account. If you request the return of an excess contribution, the employer will make the corrective distribution from the designated Roth account.
If you have pre-tax funds in the 401(k) at the previous employer, perhaps from employer matching contributions, you can do a taxable rollover of some or all of that to a Roth IRA.
If the current plan allows In-plan Roth Rollovers, simply do an in-plan Roth Rollover of the amount unintentionally contributed as pre-tax.
If you do nothing, you'll simply have an amount that will be excluded from income now (excluded from box 1 of your W-2) and will taxable later when distributed and not rolled over to another pre-tax account. If the current employer makes pre-tax matching or profit sharing contributions, these pre-tax amounts will just be combined in the pre-tax account.
