Road fixed / fixed road

There's a trick with a biro spring. Or a Problem Solvers 2:1.
EDIT: you can also (probably) tape an inner cable to the bars, taut.

I put a rubber o-ring round the strut the cable anchor goes through at the top of the brake lever, and other end round the handlebar. I originally used a rubber band but it broke after a while.

Yeah, I've tried taping down a shorten cable with electrical tape under the bar tape. It does work, until you have an "oh sh1t" moment and instinctively go to grab a fist full of brake. If you look at my winter hack I've drifted the pivot bolt on the LH brake out and removed the lever all together. That works the best as zero rattles, but I didn't want to do that on my "best" fixed gear as that was running rare Campagnolo Record single speed specific levers, hence the electrical tape trick.

I've just stuffed a load of balled up tape in mine before. Simple but effective!

Is there a best technique for preventing the rear brake lever from going all floppy when only running a front brake?

If you look closely, you'll see a wrap of self-amalgamating silicone tape around the top of the left lever here. It's rubbery, so if you do grab the lever it stretches and then pings back.
The JTek cable doubler is supposed to be better than the Problem Solvers, and it would only need to be half as good to be the same value for money.

Yep, putting a cable and clamping it on the bars works a wonder.