I don't think it's grounds. The heat means that there is an excessive voltage drop at the fuse block.
First question, what are you using for a fuse? Lucas repro or AGC?
Assuming you have either the proper Lucas "35 amp" or an AGC 20, then the problem has to be a poor connection at the fuse block. I'd start by polishing up the surfaces of the tabs that grip the fuse, with a Scotchbrite or similar, as well as the mating surfaces on the fuse. If that doesn't solve the problem, then you'll need to do some checking with a voltmeter or DMM, right at the fuse block, with the heater blower running. You should have no more than about 0.1 volts from the white wire (which is power in from the ignition switch) to the green wire (power out to the heater, turn signals, etc.)
I forget offhand how the TR6 fuse block is constructed; but on a Stag fuse block, there are copper rivets that are supposed to carry the current between the fuse clips and the wire terminals. Over time, the plastic shrinks until the rivets are no longer tight, and then the copper corrodes to provide a high resistance connection.
FWIW, I once had a similar problem with the (aftermarket) headlight fuses on my TR3A. Couldn't see any corrosion, but the tabs were not making good contact with the fuse even after replacing the fuse. The heat literally melted the solder inside the fuse! Which led to a fuse that was visibly fine, but would not carry current (the broken joint was hidden inside the end cap). Few seconds work with a Scotchbrite and the problem never came back again.