Perhaps I missed it- was the distilled water tested with a tds meter? I'm thinking if it reads 0 then it shouldn't have anything in it..? Are there things that doesn't register to a tds meter?
Let's go with tds meter being 'very reliable at detecting things in water' until further information comes along. Then this would mean the LFS was giving bad info? I could believe that perhaps some brands of distilled water either don't do it properly and/or add things to the water after the process so in the end, it's not really distilled despite the label. There's distilled water for babies, for example- the ones from Walmart at least has a very distinct smell.
From personal experience: tap water registers 150-160 ish on a cheap tds meter. Three stage ro unit(got this before going into reefing) registered 3-6 tds. I always got the generic cheap distilled water from grocery stores- several different brands all registered 0.
Here's something I thought that was very interesting... the ro/di water from three different LFS ALL registered something. I forget the exact readings- they were in the 80-100s, even the one from a shop specializing in corals and saltwater fish only. One shop's water registered something like 185- "worse" than my own tap water. I did not point it out to the owners of those shops- two are highly opinionated with a tendency to become hostile if they perceive someone to be even as much as 'questioning them'. So I won't find out what exactly was going on with their water.
Based on my own experience, I'd test the LFS distilled water also. Also due to above I just went ahead and got a 4 stage RO/DI unit with 0tds output. I see getting a RO/DI is not really an option for OP, just mainly suggesting to check the distilled water from any source to help determine if that's a potential source of the issues or not.