You are correct, code H is required for the rollover from the Roth 401(k) to the Roth IRA.
Code BG is only to be used for a rollover from a Roth 401(k) to another Roth 401(k).
The instructions for Form 1099-R explicitly state:
Note. Do not use Code G for a direct rollover from a designated Roth account to a Roth IRA. Use Code H.
and:
Use Code B for a distribution from a designated Roth account. But use Code E for a section 415 distribution under EPCRS (see Code E) or Code H for a direct rollover to a Roth IRA.
https://www.irs.gov/pub/irs-prior/i1099r--2023.pdf
