I think this needs to be handled with some patience and diplomacy.
Clearly dinner with mum and dad once a week for the childs benefit is no bad thing. I get that it is for the child.
Not knowing her shifts in advance is just a bit of a power thing.
TBH you have a partner who is a regular active part of his childs life which is a bonus. It shows you he's a good man and committed to parenting.
The 'never knowing what night we can have a regular night out' is irritating, but resolvable.
As your husband to speak with his ex AT DINNER every week. Make sure he HETS the shift pattern so you can plan around this.
After a year/ 18 months both he and you know if this is a LTR thing, or just long term casual. I know exactly how you feel as I was in the same situation myself.
As long as he is committed to you and building a future with you and encouraging a relationship with you and his child, then you are just gonna have to Suck it up until the child leaves for Uni.
If.you want him to put you and your relationship before his child, that isnt going to happen.
The issue comes when him/ his ex/ his daughter dynamic automatically excludes you from birthday parties/ dance recitals/ parents evenings etc.
At some point the ex must.make room so that you can ALL take part in this childs life going forwards, or you are just going to feel like the third wheel.
There is no point , if he's spineless asking him. Grow a relationship with this women ( his ex) Get involved in step parenting. Ring her. Talk to her. Meet for coffee. Then YOU ask HER, so, let me know what shifts this week as I'd like to plan something with ( daughters name)... what 's happening this week ?
Or try " I've promised ( daughters name) to take her to XYZ, next week. Can you tell me what nights she's with Us ?
Keep the daughter at the centre of it all. If your partner doesnt see you as part of his daughters life he'll keep co parenting with his ex.
Once his ex no longer sees you as a part time, and actively engaged in her daughters life, she too will come round.
You will have to play the long game here.
THEN... you can schedule time for you and your chap, based on his ex schedule.
If.you cant wait and issue.ultimatums to any of these parties ( himself or his ex)... you are in effect moving yourself out of the picture.
If you want a partner who works regular hours and has no parenting commitments, he's not the guy for.you.