Wow bingobingo - I'm in such a similar position!
Although I'm pregnant with our first, not second, (his second, though).
Much like yours, it was also his ex that split with him, supposedly she didn't know she was pregnant at the time. Whereas I believe she did know, and got out of the relationship for that reason. Either that or she was already seeing someone else (who she is still with now, 6 years later). I believe with how she's always been awkward, she possibly didn't want the baby to be my partners and 90% of the time, her lashing out is because she's annoyed with herself 
My partner has never set a foot wrong since her pregnancy and she hates that, as she has nothing real to go against him, whereas it would be easier for her if he actually did slip up.
She wouldn't allow him to any of the scans, and he didn't even know the sex of the baby until she was a few days old. She also didn't put him on the birth certificate. I always feel the need to tell this story in full because a lot of the time it's about 'deadbeat dads' when in fact a lot of the time it's the women/mums who can be spiteful and use the children as a weapon!
In the latest argument she even turned around and said (in a text message) there's a reason you're not on the birth certificate! She was referring to the fact that he was asking for more access, and she knows that he can't get any more without a court battle, and even then it's not guaranteed. He only gets to have her every fortnight for 2 nights.
His daughter has been very off since we told her we were having a baby. She refers to the situation as 'you're having a baby aren't you' or 'you're having a boy aren't you' rather than 'I'm having a baby brother'.
When my partner was talking with her a few weeks ago and mentioned the word brother, she said 'no, half brother'. Which had obviously been drilled into her by her mum, as we've never even mentioned that phrase before, and weren't planning to!
I can imagine her mum is turning her against us and the baby before he's even her. She even said that for the past month, her daughter hasn't wanted to come and see us and she has had to 'force her to'.
She also tried to tell him that she only wanted me to do the pick ups and drop offs, and not him. (How very convenient right near my due date!). I made sure he quickly squared that one up with her that it would be impossible.
Its very hard what to believe when she is still young, whereas one advantage you have is that your stepchildren are older.
Like someone else said, there is little you can actually do other than let your partner know you're completely with him on this.
If his children are at that age and aren't keen on visiting anymore then let it be. Unfortunately for them, they have an unhelpful mother, but hopefully they might break free of this one day.
As I got older into my teens, me and my brother stopped going over to stay at my dad's house (no influence, just growing up and wanted to do other things at weekends). I have always stayed in touch and am relatively close with my dad, whereas my brother is not so much. I think that's just because they are so similar and tend to clash.
So really, I just wanted you to know that you're not alone in this sort of situation! Good luck with your new baby 