I have three Warren May's.  Two have wooden peg tuners and one has mechanical tuners.  All have adjustable bridges.  What I have found is that the moveable bridges need to be angled a bit to compensate for the slightly flat string.  On one I simply made a new bridge out of a piece of chopstick and built one side up just a tad and it was spot on or at least pretty close.  I have never had much luck with Snark type tuners.  I'm sure they are fine if used properly and I'm just to numb to do it right. You might try slipping a flat toothpick under the nut closest to the melody string as that sometimes help fine tune the contraption.

On another old dulcimer that I have I noticed that when I play the melody string I can overcome the flat by moving my finger closer to the middle of the fret spacing.  I guess each of the builder's dulcimers have their own individual quirks.

I agree with Skip that it may be a problem with the slot in the nut, but also it could be a problem with the saddle/bridge as well. You may need to fill the slots and recut the slots for that one string.

Ken

"The dulcimer sings a sweet song."

That shouldn't make a difference, I use a phone app on mine the same way and it is the same readings as the tuners I have.  The slot for the melody string may be cut so the string touches on the tuner side of the nut instead of the bridge side, effectively making the distance to the 1st fret longer. 

Skip, my sister sent me a Snark SN-2 chromatic tuner a couple months back, haven't tried it out yet, going to put that on her today and see what happens. Been using the Pro Guitar Tuner app on my phone, and have the phone sitting on the dulcimer next to the melody string close to where you strum it, so maybe that's screwing up the reading

That makes it an interesting mechanical mystery if it has fretwire frets.

updated by @skip: 12/30/19 10:53:42PM

Thanks Ken, I'll tune it DAA then! Warren's response was "it is a traditional tempered scale"

Thanks Skip. I did what you said, and checked the tuning at the first fret on my old dulcimer- still waiting for the Warren May to get here. The bass D3 was bang on, as was the middle A3, but the melody d4 was flatter than it should have been, if that makes any sense...

Warren uses more or less Warren Intonation, not a perfect Just Intonation.  If he says an instrument is "better in DAA", you'll get the best results tuned in DAA!!

I am Robin, but I messaged him and am waiting on a reply. He did say that that one was better in DAA tuning, so I'm thinking that one probably is Just Intonation. But now I'm wondering about my old girl I got a few months back that I'm still not sure who made, that I've been chording on in DAd...

@david-messenger Perhaps this link may be of help: 

https://fotmd.com/strumelia/group_discuss/2594/just-vs-equal-temperament

I'm guessing you're wondering about the Warren May instrument. . . 

Use your tuner, then check the tuning at the 1st fret, if it's not close to what is expected, then probably just tuned. most all tuners available are set up default as equal temper. The space from the zero fret/nut to the 1st fret  is usually too big.

   How can you tell if an old dulcimer, all diatonic was set up for just intonation or not?