Is it decidable whether the langauge accepted by a reversal-bounded counter machine is deterministic?
I found a similar argument for context-free languages, which can be used here.
It is not decidable, since universality for non-deterministic reversal-bounded multicounter machines is undecidable, but becomes decidable for the deterministic version.

Suppose we can decide if $M$ can be determinized, and perform the determinization if it can.
Then we can decide if $L(M) = \Sigma^*$ as follows:

If $M$ cannot be determinized, then clearly it is not universal, since there is a deterministic machine accepting $\Sigma^*$.
If $M$ can be determinized, then we determinize it and test if it is universal.

This question on CS.SE give the analogous argument for CFL vs. DCFL.