Equivalent DPDA that always halts for every DPDA with empty stack language
The look-ahead property for pushdown automata is that a PDA $Z$ can answer certain questions of the form "how would PDA $X$ behave when given my stack", and then make moves based on that information. In particular, also deterministic PDA $Z$ can use this even when $X$ is nondeterministic.

You know, as I understand from the formulation of your question, this is possible because the property "when started on stack $\alpha$ PDA $X$ empties its stack" defines a regular set of stacks. The PDA look-ahead test can then be build in the original PDA.

So, to solve your problem, you have to formulate which PDA tests you would like to perform on the stack in order to avoid infinite computations. 

I do not know whether it is necessary to actually construct the regular language of stacks that gives a positive answer.