No, when it comes to animals it's only the rules of the operating airline that apply. If you book a ticket for a codeshare flight on the United website, even if it has a United flight number, if it's operated by Lufthansa it's Lufthansa's rules and fees that dictate if and for how much animals fly.
Lufthansa still accepts snub-nosed breeds as cargo (for crazy money), but they stopped accepting them as checked baggage in 2020: https://www.lufthansa.com/us/en/dangerous-dogs.

If you don't want to drive from Chicago to Tulsa, you can also fly from Malta to Rome on ITA, then fly from there to Dallas on Finnair via Helsinki. Finnair does not ban snub-nosed reeds from the cargo hold, and they fly to Dallas (DFW) which is just four hours from Tulsa. You'd need to book this on separate tickets, though.
Finnair used to be my go-to airline with pets in the hold, until they decided to triple the price for this service. It used to be EUR 200 for a dog to fly to the US in the hold until 2019... now it's EUR 600, but they do allow two dogs in one crate, so that eases the pain in your case.