Hiya,
Yup, you've encountered a known limit of the lambda functionality. The two options you outline are the ones I'd consider. You could update the user data with the tenant name on create using a webhook, so maybe not as messy as you might think.
The only other option would be to file a github issue requesting the tenant information be made available in the lambda: https://github.com/fusionauth/fusionauth-issues/issues
There are some similar issues I'd suggest voting up if this is important to you:
https://github.com/FusionAuth/fusionauth-issues/issues/571
https://github.com/FusionAuth/fusionauth-issues/issues/267
https://github.com/FusionAuth/fusionauth-issues/issues/229