I can't seem to find the correct ombudsman in the UK. If you can recommend one I'd really appreciate it.
The ones linked from the UK gov's website result in channelling us to Apple support, or state that Apple don't cooperate with that resolution service. eg ..
I'd imagine the consumer ombudsman, however your best bet is to go to Citizens Advice if you feel you have a genuine complaint.
As to the resolution I think you should be realistic, whether you take that advice or not doesn't bother me. At best you'll get a refund, more likely you'll get an apology, at worst you'll get nothing but tens of hours wasted and stressed out with everything.
At the end of the day the computer performs as expected, it is only that your expectations are higher. Apple have made attempts to correct any faults and seemingly have performed a lot of repairs on your computer, so given the financial cost that they have incurred in attempting to resolve the issue and the subjective complaint, it is unlikely you would be able to prove your outcome to your satisfaction. If it was a company that just shut the door on you and did nothing you are in a far stronger position to argue.
As I said earlier you would need verifiable independent evidence that backed up your claim, and you need to be very specific about your claim. Simply saying "It runs hot" is not a complaint, it's a subjective opinion. If you had another identical unit which did not do the same, then you can say there is something wrong with yours, but it needs to be identical in every way and tested under the same circumstance. If you feel it should not get warm at all then you need to prove why, suggesting your phone/tablet does it without warming is not enough as it's different technology. You would need a very similar computer to run the tests.
However crucially in all of this is the feeling that it getting hot is subjective, it is not malfunctioning and is not overheating. It is operating within stated parameters and so there is nothing misleading occurring with stated functions. If you tried to complain about any of this, it'll just get thrown straight back at you.
Collecting information on an internet forum is one place to start. But I would be concerned that all you're doing is validating your viewpoint by ignoring opinions that differ to yours, this is just further entrenching you into your belief. I'd guess it's becoming quite stressful not getting anywhere, going around in circles, and yet being fully adamant you are right.
Take or leave the advice but it'd be far healthier to just drop it entirely and either enjoy your computer, or sell it and buy a different one. I say this purely because I personally can't see an end that will fully satisfy you on this matter (The longer this goes on the higher your expectations are becoming), at least not without spending money on collecting independent evidence or court fees, you'll end up spending a lot more than you would if you just sell it now and move on.