So I know that this might sound a bit back alley, but what about non toxic liquid soap? 1/2 drop per liter and I can't taste it in root beer, but the foam stays.
Ultimately you need a surfactant like yucca root extract(I couldn't find this anywhere), but that is what makes the bubbles stay not under carbonation. Think suds bucket, has no carbonation.
Try different amounts in a pint of water whisk a bit look at suds, then taste. Us the least amount possible then scale it up. If you force carb, the carbonation is there it just is letting the bubbles pop too easily.