Changing the code-H Form 1099-R is not an option.
This is a serious error because this distribution from the designated Roth account was not eligible for rollover to a traditional IRA. The deposit is a failed rollover and therefore constitutes a regular excess contribution to the traditional IRA, subject to penalty unless corrected. The brokerage was clearly not very helpful in suggesting that this could be resolved by a tax advisor.
The excess contribution must be returned to you from the traditional IRA by the due date of your tax return, including extensions. I would certainly request a filing extension by April 18, 2023 to ensure that that deadline is extended. You'll need to convince the brokerage that this is indeed an excess regular contribution that is eligible for return. In processing a return of contribution, the custodian should calculate the net income attributable to the excess contribution and distribute the adjusted amount to you.
As for getting the money into the Roth IRA, you need to get the brokerage (or any other Roth IRA custodian) to accept self-certification under IRS Rev. Proc. 2020-46 (reason 3.02(a), financial institution error, and 3.02(c), the distribution was deposited into and remained in an account that the taxpayer mistakenly thought was an eligible retirement plan, would seem to apply) that these circumstances would qualify for a waiver of the 60-day rollover deadline to allow the rollover to the Roth IRA of the original amount distributed from the designated Roth account in the employer plan: https://www.irs.gov/pub/irs-drop/rp-20-46.pdf
Once all of this is done, you'll be able to report the code-H Form 1099-R as properly indicating a rollover to a Roth IRA (despite the fact that the rollover became an indirect rollover). If there are any investment gains attributable to the excess contribution to the traditional IRA, the gains will be taxable on your 2022 tax return but you will not receive the code-P Form 1099-R reporting this until the end of January 2024. Whether there is a gain or a loss, you'll need to include an explanation statement with your 2022 tax return describing the return of contribution.
If this was a new traditional IRA account containing nothing but this excess contribution and it's attributable net income, the custodian will not have to do any special calculation to determine the net income attributable to the contribution being returned and will simply distribute the entire balance of the account. The distribution should not be done as a regular contribution. (Additional steps involving another Roth IRA custodian might be necessary if the brokerage currently holding the accounts is uncooperative.)
