I'm really nervous about any suggestion of a constant-time flag for LLVM for two reasons:
1. LLVM does not have a notion of time in its abstract machine, so you are introducing an entirely new concept but applying it in a single place. The C abstract machine does not either. One of the most common flaws in papers about constant-time extensions to C is the assumption that they are preserving an invariant, not adding a new invariant.
2. Most ISAs do not make timing guarantees about individual instructions, so even if we define and preserve the guarantee throughout the optimisation pipeline, we can't guarantee it in the final binary.
We discussed some of this on our paper on preserving security invariants through a compiler pipeline a few years ago.
In the second case, consider something as simple as an add. There have been ARM implementations where a 32-bit integer add took one or two cycles depending on the operand values. To guarantee constant-time execution, you'd need to ensure that you toggled some bits in your values to guarantee the slow path and then reassembled the result. That's a big codegen effort, but is required only for one or two our of hundreds of microarchitectural implementations of the AArch32 ISA.
More troubling for most people should be the fact that Intel *explicitly* does not guarantee that CMOV is constant time. There are possible microarchitectural implementations of this instruction that handle it entirely in the scheduler so that the instruction commits as soon as both the condition code and the used operand are available (side-effect-free instructions that lead to the unused value can be silently dropped). No one implements CMOV like this, but Intel is not willing to commit to *never* implementing CMOV like this, so any code that assumes that using select instead of branches is constant time is not guaranteed to be.
The lack of any kind of notion of time in the LLVM abstract machine makes plumbing this in very difficult. Informally, a constant-time abstract machine has no data-dependent flow control and has no operations that have operand-dependent timing. In most ISAs, this eliminates data-dependent loads and stores, floating point instructions, and integer division.
If you want to support constant-time execution, I'd recommend defining markers for the start and end of constant-time blocks (e.g. crypto kernels) and explicitly annotate input values that are not secret dependent with metadata. Then ensure that this metadata (insecure values and safe input values) is emitted in the resulting assembly. You can then write a per-ISA (probably tuned per-microarchitecture) validator that ensures that the output is constant time according to your model. You can then work backwards to find places where LLVM does transforms that are breaking your assumptions and work on patches to fix them.