I hope you had the happiest of birthdays, in spite of your H's stupid selfishness. I know how difficult compartmentalizing these things can be.
This is one thing that will always get my goat, try as I might. I just cannot abide someone exposing their children to an adulterous partner -- I am sorry if that offends anyone, but that's just the cut of my sails -- and I am particularly sensitive to this because of my own case.
But yes, you will all ways be DDs' mother. H might stupid and foolish enough to try, but he can never take that away.