An update to my original post: We ended up purchasing a small bathroom faucet for our RTrek kitchen sink. It wasn't too expensive so we figured we would try it and see what happens. Even though the new faucet doesn't fold down into the sink like the original, it does work just fine, and fitted the sink hole which had been our concern.
Thanks again everyone for your help!
https://smile.amazon.com/gp/product/...?ie=UTF8&psc=1.