Because you had requested a trustee-to-trustee transfer, I assumed that the "new IRA" was established as an inherited IRA. If not, this is truly a taxable distribution from the old inherited IRA and a massive excess contribution to the new IRA. In that case the new IRA custodian did nothing wrong in thinking that this was a rollover from your own IRA and allowing the deposit of a rollover, but the deposit still does not legally constitute a rollover.
Whether the account is titled as inherited or not, it does not constitute an inherited IRA and the deposit does not constitute a rollover. Instead, the deposit constitutes a regular contribution, far in excess of the contribution limit from 2022 of $6,000 (plus $1,000 if you are over age 50). With a roughly $244,000 excess contribution ($250,000 minus $6,000), the penalty is 6%, roughly $14,640, every year that the excess remains in the account.
