You do not mention what kind of rims you are using, which may matter a lot with regard to the braking that you have on the record brakes. Assuming a quality aluminum braking surface, housing, and pads, I have not experienced a significant difference between the record brakes and mechanical discs in terms of dry braking power. The discs probably will, however, have better heat management, which is what you are running into here. So I imagine the overall braking would be better with the front disc. And is the record rear a single pivot? Might not hurt to change to a dual privot caliper as well. Will not help with the heat, but might let you cycle the brake on and off a little more which could let it cool. Something else to consider is a deep section rim. A deep section rim has a lot more surface area that would move heat away from the brakes and cool faster than a box section rim.
I have taken the Spyre on Alpine descents, and been fairly happy with them. Have used the HY/RD, but not for serious descending. I do not find a huge difference between the two in regular use. I've had a TRP rep tell me that the HY/RDs have no problem with heat build up, but, personally, I'd probably put the Spyres on, There can't be much fluid in the HY/RDs and it just seems that you would boil it pretty quickly. I've looked for finned brake pads for these brakes, but haven't been successful at finding any.
No problem running 28 front and 25 rear, though you might find the front disc works well enough to avoid the problem with the rear brake.
You'd need to modify the frame to get a rear disc...might be worth checking with Cyvac. Calfee is going to do it for our carbon tandem. Not cheap, but cheaper than a new tandem.
Where are you riding? I'm assuming France given the Cyfac brand?