All flex working requests you should have a meeting to discuss why you need it/ put forward your request formally then they have a give time to consider you're request then a follow up meeting for their decision. You can have someone with you in these meetings I didn't I went alone thinking my request was straight forward but regret it now. If they decline they need to have a proper reason i.e. how it would impact the business they just have to show they have considered it. Sorry you are in a toxic job it sucks doesn't it
Its quite a standard practice to hold meetings for flexible requests. The business only needs to agree with that if it suits the needs of the business. So it might be they need a bit of give and take or they dont have capacity for you to take the day you are requesting. Go to the meeting and see. They can't bully you out of a role Have a look at pregnant and screwed. They would be able to support you