I'd tend to go RODI, since if you have a problem, it can end up costing you more in specimans than the RODI unit will.
We had our local water tested down in Oklahoma and found it contained a plastic compound which is quite bad, and arsenic, from the ground water. We tested distilled water we bought, and we found out that the company (a well-reputed company) was not careful in preventing what they distilled out from running back in during the bottling process; and on another occasion, we had a gasoline smell from a bottle of delivered drinking water: they had taken a bottle back from a customer that had been used, probably, to hold lawnmower fuel.