FWIW, if you're the administrator of the organization, you can disable 2FA from the admin console for that user's next login. I've done this a few times for similar reasons.
The difficulties were to be expected as personal Gmails weren't meant to be used like this (the goal was just to share an anecdote about the difficulties of phone numbers used for two-factor authentication with the free service even once a year). The long-term solution we used was to pay for a reliable but low-cost (in comparison to Outlook and Google) email host initially recommended on HN and a few sysadmin forums, to gain access to organization-wide admin features.
[1] https://support.google.com/a/answer/2537800?hl=en#zippy=%2Cc...