I use red boat trailer bearing grease.
Check that it's compatible with the seals. The whole DoT/Mineral/LHM/whatever issue applies to grease on the other side of the seal as well.
Copaslip does a pretty good job and doesn't affect either type of seal.
And you can probably exceed the liquification point of most greases in a braking system, so you'll get runny grease exactly where you don't want it.
Copaslip is "better" but you may still get some of the components melting, not sure what the make up is of most copaslips, the one i have now pretty much drys out after 24 hours and is good to several hundred degrees, no greasiness left, the last stuff i had was actually a copper loaded grease, so stayed greasy forever and you'd get the grease running out at high temps (but leaving the solids behind).
If that makes sense.