TBH, they are technically not far off following NICE guidelines on the subject.
The guidelines for maternal requests are very much about psychological reasons and the advice to offer counselling.
Virtually all hospitals will try and put you down this route - and ignore the significance of the word 'offer' (meaning you can refuse) part of counselling advice. This is because hospitals are not obliged to follow the guidelines (some are actively ignoring it completely and have blanket bans on maternal requests) and are interpreting them in various different ways.
There is actually no reason to get offended by the suggestion of a psychiatrist - in fact I think you are being slightly wrong in your attitude by getting angry about it in the way you are. Hospitals think they are following the guidance (and justifying a maternal request CS in the correct manner to their superiors due to politics of CSs) by doing this.
I think changing hospital you may therefore may well come across the same issue. It really depends on which individual consultant you see, and their attitude to the whole subject. The truth of the matter is there is such disparity of care with regard to ELCS, that the situation is it can equally be an individual consultant that is the problem or the hospital policy and from a patient's point of view it can be virtually impossible to tell which is the obstacle as there is little transparency on the subject.
The important thing to know is that the guidance also says you are entitled to a second opinion if a first consultant refuses to perform a CS. At 30 weeks you have some time but not much to get this resolved. Changing hospitals may in fact make it more difficult because you aren't 'in the system' with them and it may take time to see a consultant and then there are no guarantees they will tell you any different. In someways it may be better to stick where you are and either go along with the referral and argue your case through that, or to ask to see another consultant.
I feel for you, as there is no easy solution to this, but I do think you have to understand the politics that are going on here. Politics that are not interested in women but are based on ignorance, pressure to reduce CS rates and guidelines that aren't properly understand in why they were introduced in the first place and the reasoning behind them. There are some consultants who are being really difficult and indeed unprofessional about the whole subject. It shouldn't be like this, but it is.
Good luck it trying to get it sorted though. I've seen only two MNetters who ultimately don't get an ELCS if they really wanted one. Many have had to really battle to get this though.