Is the halting problem always decidable for non-universal programs?
To answer the first part of your question - Two counter machines are not exactly Turing-complete. That is, there are some functions that cannot be computed with them, for certain counter-values. This is a very slight weakness, so 2CM are actually very close to being Turing complete, but still.

As for your suggested model - could you formalize it in more detail? It's not clear enough as it is.