I think I found the very long way around to sync Garmin Index weight to Zwift using FitnessSyncer (
https://www.fitnesssyncer.com/) - I needed to recreate my Fitbit account, link Fitbit and Garmin Connect to FitnessSyncer. Than connect FitBit to my Zwift account. Here's hoping that once it set up, I can forget about it and it'll just work.
Not exactly a user friendly set up ... blarg. And I've been trying to deactivate as many accounts as possible to minimize exposure to hackers, so creating two new accounts is annoying.
Funny enough, I remember doing something like this 5-6 years ago to automatically sync Garmin rides to Strava.