You've already had quite a few suggestions on the way to help. I hope I can add a few more from a local perspective. (I'm in Gloucestershire, and as well as knowing what 'a stone' is*, I do know a wee bit about the nhs.

1. I definitely think your case should be reviewed by one of the consultants. At Gloucestershire Royal, the sleep medicine department has grown, thanks to case-load, and is now almost 'stand-alone'. It's still officially part of Lung Function - but most patients only see the consultant once after the sleep-study, and thereafter it's one of the sleep physiologists or one of the specialist nurses.
Given that apnoea which is confined to REM sleep is relatively rare, and given your other recent medical history, I think another sleep study - one in the sleep lab - is justified. And, as I understand it, only a consultant can request that.
2. I note what 'the technician' said about a three-month wait to see the doctor again. I would have thought that, you being a medical professional, they might find their way to seeing you sooner. If they need a justification, it's worth remembering the NICE guidelines which say: if treating your sleep apnoea is critical to your job, you should be fast-tracked.
3. From what you say, your foundation trust hospital has standardised on Philips Respironics machines (in most of the UK, it's either them or the other major manufacturer, ResMed), and they've given you an auto-adjusting one - model name 'Dreamstation'.
That's a good sign - in some parts of the UK, the standard issue is still a fixed-pressure model - either from the Dreamstation range, or the previous generation of models.
On the whole, people do better on APAPs.
Having said that, your REM apnoea might be better treated overall with an auto-adjusting bi-level machine. Philips Respironics do make one, but since bi-levels cost approx 60 pc more than APAPs, an 'individual funding request' would most probably be needed.
Another reason why you should be seen (and soon) by the consultant.

4. You mention that they've run out of humidifiers. I get that you're not convinced it would make much difference - but you'd be surprised. It isn't so much the increase in humidity (not in our island nation), it's the temperature.
You might ask if the sleep medicine department can just loan you one for a week - to see what it feels like. And if they still can't issue you with one permanently, there's nothing to stop you buying your own humidifier - which you can do direct from Philips Respironics UK. Cost about £120. They're in Chicester, and the phone number (within the UK) is 0800 130 0845.
5. You've done one thing right - you've found this forum. I'm a visitor here myself - but I can say the level of expertise is remarkable. I routinely pass on the web-address to my specialist nurses at my annual review - as a suggestion for their continued professional development.
* A 'stone' is a unit of weight measurement. 1 stone = 14 lbs.