Just to clarify, as I understand this question, no change in the equilibrium will occur, but a catalyst may change the kinetics of the reaction. The problem is the low solubility of the reagents in a given solvent. If water or alcohol were used, the solubility and reactivity would change and the reaction would occur, though with an intermediate reaction of an ROH with NaH.
If one were trying to react the hydrochloride salt with NaH in a solvent like toluene, the low solubility would make this reaction very slow or no reaction at all. If a catalytic amount of t-butanol were added, it could react more quickly with the NaH as it is soluble in toluene. The NaOtBu could react more quickly with the amine-HCl if its solubility were similar to KOtBu, and regenerate t-butanol.