How about making operations constant time in application code by picking an upper bound which is acceptable for the application, but that is certainly longer than the actual CPU computation and then waiting until the upper bound to return the result?

Eg. my app is performing digital signatures and I'm sure that they take <1ms CPU time, but performing digital signatures in 10ms is acceptable for my application, so when I perform a signature I measure the CPU time elapsed, say 0.5ms and then wait for 9.5ms.