I suspect it's a combination.
They probably have very little experience, though that will be because they refuse to attend training sessions.
They probably believe, that the expense of referring patients is unjustified. That the problem is largely vanity. That most of 'these people' will simply go away if they are ignored. And they can save money.
There are probably quite a number of other people with gender identity problems, who have already been turned away.
You need information on the realities of transgender. That means statistics and not simply anecdotal references and defiantely nothing American. (Sorry to our American friends, but their health service is not taken particularly seriously in these matters). There are many sources in the UK. But web references are not usually much help since these people are trying to be obstinate, they can still claim they don't use the web. Total lie of course, but if honesty were important, there would be no need for checks.
So, you need stuff you can cite. NHS resources are mostly online, though rather patronising at times. These can be printed out and are very legitimate. Also, try can contact some transgender groups in your area, the local LGBT organisation should know.
You will need to remain very calm. If you get fed up, if you get depressed. If you feel you just want to get a huge gun and blast them all to whatever, walk away. This is a matter for cool head.
If you decide to run off and try another health board, then you will be starting from scratch, you have no guarantee it will be any different and you will be letting down your transgender brothers and sisters, each of who has an absolute right to be treated like humans.