You might think that anima in the reservoir of your first covenant is ‘lost’ upon changing covenants, as its stored in your old covenant’s reservoir and would be available if you ever returned there.
You can however bring it with you quite easilly!
Lets say you have 3000 anima in the reservoir of your first covenant. Buy x4 of the covenant hearthstone and move to activate your new covenant that you havent joined previously, or turning in the second prove your worth quest.
Upon arriving, and within the buyback time limit, return all 4 hearthstones. And voila, 3000 anima to deposit in the tank of your new home!
I know it works, Ive done it myself when moving from Night Fae to Venthyr.