The Withings API is read-only, so you wouldn’t be able to sync anything to Withings, even if Basis had an API…
Still looking for a good solution for continuous heart rate monitoring. But for doing quick resting heart rate tests, Azumio’s Instant Heart Rate app seems to work quite well, certainly better than the Withings Pulse. You can send the data to 3rd party services via Google Fit, if you’re on Android.