You can spend time spinning your wheels trying to find replacement handles that likely will cost you $25 plus shipping or you can just bite the bullet and buy a replacement faucet. If you choose to get a replacement "Lasalle Bristol" faucet, you can find one on Amazon for about $37.
https://www.amazon.com/LaSalle-Brist...aSalle+Bristol
If you do choose to do that, you'll get the same kind of "inferior RV quality" faucet that you're trying to find parts for now.
The alternative is to simply go to Lowe's or Home Depot and find a quality kitchen faucet and install it in your RV. They are pretty much a "one for one" replacement. Then, in the future, if you have problems, you've got a "lifetime warranty from a local store"......