Hi Martin
It definitely sounds like the one channel of the Key-Less is tripping off - when you turn system off and back on again it resets - so that part I understand, at least as far it re-setting with the off/on cycle - however why it should come back on again when you touch the brake is just weird!
I can't see any possible explanation for that.(edit - read on!!!!)
As per that circuit tripping - it is only supplying relatively small current to three circuits which explains your headlight and ABS issues at same time (and also actually your running light aka park light - if you were to check that in your fault condition, that should also be off). Any short or excess current on that channel will trip the device (just that channel) and it will NOT come back on, even if short no longer there, until you turn whole thing off and on again (i.e. Kill then Run + new Clutch Pull)
Directly off the Key-Less it feeds into the ABS controller and via fuse # 8 it feeds the enable for the headlight relay and also loops back around through the key-switch connector (not actually into the key-less box, just a loop on the connector) and through fuse 10 to the park lights.
As I'm typing I think I figured it out - I was about to state what possibilities were for the circuit to trip while disregarding for a minute why the headlight comes back on again when you press the brake, when I realized a possible scenario that would cause both the trip AND the apparent re-enable with the brake:
Check the tail light harness under the fender - those wires often rub on the tire and through the insulation exposing the cores; if the running light feed shorted to the ground feed, that would trip the key-less circuit; then when you operate the brake, it shorts to the running light wire and turns the headlight on (important distinction here - this is NOT the key-less channel turning back on that is enabling the headlight, it's the momentary connection from the brake wire touching the running wire and why it goes off again when you release the brake). I could have sworn we already had this issue with you did we not, or am I thinking of someone else?

The other possibility could be in the tail light bulb itself (if its incandescent) - I've seen blown/broken filament before touching the other one inside the globe! But I think the harness is more likely, it's far from uncommon problem.
My first thought re the circuit tripping off was that it was another common issue where the headlights have been replaced with LED units, and the park light bulbs are pulled and the sockets actually short out without a bulb in place (Maybe that is the previous issue on yours we had I'm thinking of?) - I was just about to write that first and thought 'OK, but that doesn't explain the brake light thing' and then had a eureka moment (well I hope it is!)
Check out that tail light harness and see if all looks OK there - I would also suspect that fuse 10 could be blown, if your park light is not on 'normal' without applying the brake and prior to the headlights going out.
p.s. I just caught this post by chance - use the @ symbol before my name and I will get a direct alert about the post. e.g.
@Martin-Brighton-uk