He has an issue. And it’s up to him to get it addressed. Obviously, there is an issue with lying which is a way to cover it up. In his mind tell a small white lie to cover up the greater problem .

It’s clear what you need to do. You can love someone but how far can you support them if they choose to continue deviant behavior? And if he gets caught? Are you willing to continue the facade ?

It’s perfectly fine to break up with someone you love if that’s what’s in the best interests of them or both. You will languish over the dilemma. But unless you give him an ultimatum of get help or I’m out, is he really going to get better? Also, lies multiply. And they get easier to tell and they become truth to the person telling them. So much so they believe them and are unable to discern actual truth.

If you are disgusted by his past/ current behavior , I think that gives you the ethical right to break up