What is it about lying that makes you feel gross?
If you're feeling guilty because you're keeping secrets, how does it benefit your partner to relieve your burden of guilt on him, making him carry the burden of your dalliances?
Another perspective... if you're feeling so gross about sneaking around, why is your first solution to tell your partner? What about ending the flirtations as a way to stop sneaking around?
I'm fortunate not to be the type of person who's burdened by guilt, I work more on a "will this harm or hurt another person?" mentality. If something is harmful to someone I love, I probably won't do it. If something is harmless, then I see it as privacy not secrecy. Working, of course, within the boundaries of mutual agreements of disclosure with anyone relevant.
In your case, it's the telling not the doing that will harm your partner, according to his own statement of preference. Perhaps your psyche is affected by the monogamous culture in which you were raised, where "having a romance with someone when your partner doesn't know about it" is cheating. But it's only cheating if your partner isn't given the opportunity to know. If they've been given disclosure and have requested non-disclosure, then it's not cheating.
“As I am sure any cat owner will be able to tell you,
someone else putting you in a box is entirely different
from getting into a box yourself.” —bisexualbaker