You can certainly play a just intonated dulcimer with other instruments, particularly if they are providing backing such as guitar chords. If you are having problems with things sounding 'off' then it is more than likely the instruments are not tuned correctly to blend. It is true that some specific notes of the scale between the instruments will be slightly different but if you are not both sustaining that particular note in isolation then you are not going to hear the differences. I play old dulcimers all the time with other instruments and JI really is not a problem - however, generally, tuning is an issue when playing any two or more instruments together. And that's where you should spend some time, getting the instruments in tune with each other.
In any bluegrass string band you have some instruments in equal temperament (guitar, banjo, mandolin) and some in just intonation or Pythagorean 5ths (dobro, violin, bass) and then the voices will be in natural intonation for the harmonies. So it really does all work together OK.
The 'problem' with JI on dulcimers is that DAd chording doesn't work for some chords; the 1,0,1 A chord is a particular problem as in DAd the first fret is played as the 2nd of the scale whereas in DAA the first fret is played as the 6th of the scale.
There probably was a couple of ways early builders set the frets using another string as a reference. The first was to have the bass string as the root note as in DAA with the melody string a 5th above - this will lead to the first fret being closer to the nut than it is to the second. The other system could have been to use two strings in unison as the reference as in D,d,d so the frets were set against a 5th drone. This will lead to the 1st fret being positioned a little sharper although still being the 6th of the scale. And, yes, there are in fact two potential positions for the 6th of the scale when set by ear, both of which sound OK. In fact, the classical Indian just intonation diatonic scale give the option of either position for the 6th. So you will see some old dulcimers with what looks like a more 'normal' position for the first fret even though the dulcimer was built for playing the scale from the 3rd fret rather than the nut. In these cases it is likely that the maker used two strings tuned in unison when setting the frets by ear. Now I can't prove any of this as being exactly how early makers set their frets but I can say that if you use these techniques you will end up with fret placements that match a number of old dulcimers!
For the Heritage dulcimer project I used the later Leonard Glen fret pattern. He actually has two different 6ths on his dulcimers. The first fret in slightly sharper than Ptolemy's calculations but the 11th fret is just intonated. This gives a shade more flexibility of tunings and, as you are never going to play those two frets together, having two different notes for the 6th of the scale doesn't matter.
The bottom line is that most folks are not going to hear the difference between equal temperament and just intonation when instruments play together. But you really will notice the sourness if you try to play an older Warren May in DAd. However, playing a Warren May in DAA with other dulcimers in DAd should not be an issue as I can guarantee that any group of dulcimer players will not all be so accurately in tune that you'd be able to specifically hear the very slight difference of the just intonated notes in the general cacophony!!!