Getting the user to use their app is pretty important. You may only be using TOTP now, but it allows for more intelligent multi factor authentication later on.
E.g. the app could check your risk profile, like where you're accessing from and if any impossible travel took place. They may add multi step auth like push notifications or biometrics.
By letting customers use Google authenticator you are limiting MFA to only TOTP. MFA isn't just an on and off switch anymore.
Some vendors would allow you to skin their SDK to essentially have your own version of their app published, but that is a lot of work and has its own security risks.
There isn't really a BYO app that gives you the functionality a vendor app can give.
Good MFA is harder than people think.