With the SR-71, I mostly climb with the autopilot. Granted, I
usually set at the default 1,100 fpm climb rate, but I didn't think
that would be too much of a problem for the fastest (known) plane in
the world, a plane that pulls Mach 3.3 at 80,000 feet. At some point
well before 80,000 feet, the autopilot more or less fails, pulling the
nose up into a stall.

is that autopilot capable of maintaining a certain IAS during a climb?
if so, i'd try 300 knots or so and see how it goes. if not, try keeping
an eye on the IAS and lowering the 1100 fpm if IAS goes below those 300
knots...