Joseph, personally I do tilt the dulcimer up a little bit, not fully vertical like a guitar, but just enough so the back is off my lap. But I do that more for the comfort of my strumming hand and not my fretting hand.
First, most of the time when dulcimer player refer to barre chords they do not actually mean they finger them as barre chords with a single finger. Most dulcimer players fret across all the strings on the same fret by using either their middle, ring, and pinky fingers or their index, middle, and ring fingers. Mark Gilston does the former; Linda Brockinton does the latter. I do both, depending on the chords or notes immediately before and after the barre chord.
Second, people who do barre a lot with a single finger tilt their hand so that it is more the side of their finger making the barre than the soft bottom. Using the side allows you to use the bone of your finger as a brace. If I am not mistaken, Stephen Seifert uses his ring finger to barre in that fashion.
Third, professionals who barre a lot get dulcimers with what is called a radiused fretboard. That means the fretboard is curved so as to allow an easier barre. Almost all guitar fretboards are radiused, but a different degree of curve is necessary for the dulcimer. Both Aaron O'Rourke and Erin Rogers play on dulcimers made by David Beede with radiused fretboards.
Fourth, Ken is correct that as you stretch for more chords you will be using muscles in your fingers and hands that have rarely been used before. Some discomfort at first is to be expected. But you should definitely not continue doing things that cause you pain. In order to play well you have to be comfortable. If some chords necessitate a stretch that hurts, find another way of playing the chord. Over time those stretches will be easier and easier, but you certainly don't want to hurt yourself.
Fifth, Ken is also correct (notice a pattern here?) about arched fingers. Obviously he is not referring to forming a barre chord with a single finger, but when you finger the fretboard, you should be using the very tips of your fingers, not the fat pads. To do that well might require developing some strength in the fingers, but that will come with time. When my daughter plays the piano, I encourage her to hold her hands so that an egg could fit beneath her fingers and palm on the keyboard and her hand won't crush it. As a general rule that is good advice for the playing the dulcimer as well.