OK the BB aside, the spineless twunt aside, one thing is ocurring to me.
If this were a normal relationship, ie that the OP and her H split previously, no-one else involved etc... HOW long a period would we suggest that the new relationship(s) need to go for before we introduce new partners to them....
WAY MORE THAN 5m.....
Solost, You have to hold your head up, draw yourself up to your full height and explain this to H, that it is too soon for the DC to be introduced to the BB, that he can't possibly be sure of a relationship so soon, it could all implode tomorrow and explain your rationale.
Course if it happens to sew some seeds of doubt in his head that the relationship may not last.....
I disagree with the text replies, it lowers Solost to the BB level. Don't engage with her, don't give her the satisfaction of her knowing you are riled. Change your number if she texts you again. Keep doing so every time she does and tell your DH to buy a muzzle if she really won't stop. Look into legal solutions as a last resort. Likewise if she conspires to engineer a meeting, go legal. No emotion, no anger, but cold hard legal.
Dignity first and foremost, imagine you are going to have to report every word to the DC.. what would they think if you rant and rave at their daddy? I recall my mum calling my dad a bastard once, it hurt, even though I hated him for leaving...
BB is such a ffing flake that she will be her own undoing, but you Solost have to be the picture of calm, of cool and of general bemusement at her shenanigans.
Simple, to the point and consistent.
" all access to the DC will be organised strictly between you and me, DH"
"She's your problem, you have to make it work, explain it to her, I'm not going to be involved, nor in contact with her, She is nothing to me, I can live my life without her"
the odd 'Do you really have to bend, scrape and ask permission to see your own DC???' Incredulous, Anyone who stated Me or the DC, wouldn't touch the floor on their way out the door.....