I think you're moving up two different problems.
Things that aren't safe for you and don't accommodate your pregnancy - you need to refuse, politely, with medical back-up.
Things that aren't appealing but are safe in pregnancy, like moving around, short term stints in other departments - you aren't entitled to refuse these things because of your pregnancy.
Now, if you have clinically diagnosed anxiety, whether related to the pregnancy, work, or anything else, that's another condition again. GP could document it.
But, be careful. You can't back them into a corner until their only option is your ideal job. They can say no to that.
When you say you know things won't work because they won't be well managed, you are creating a problem. If arrangements work on paper, you need to give them a try until they go wrong. Then you can point to problems.
Get GP to write fit for work so long as breaks every x minutes to move around, freedom to use bathroom at any time, no heavy lifting or bending (for example). Then you need to let managers do their job and allocate you work that fits the bill. You can't just tell them what you want to do. But be assertive about what you can't do, on medical advice.
Think about what you want from this employer. It sounds as if social side of work matters to you. Do you hope to stay with them and get flexible shifts later, around childcare. Are there other employers in the area for you?
Your pregnancy matters and I think this is a bit of a soup - you don't trust them but a doctor can't write a fit note adjusted to allow for that. Focus on getting a clear list of conditions from your GP. Then make them your boundaries and stick with them. That's where you need to be assertive (or decide the job isn't worth it).
Good luck, and do talk to union, ACAS, charities as people have advised. Draw a firm line on the real problems - what's safe for you in pregnancy. Don't complicate that with every other problem and preference. And when they come back with an improved offer, work with them, don't try to guess their motives. These people don't matter to you and your baby. The money does. Keep it professional and expect the same of them.