long winded but I have found the best fingering for barre it to use the little finger. This leaves the thumb and index finger to move to different frets. Inversions are simplified using the thumb for instance to go to the melody string to the bass string with ease. Using the ring middle and index to barre leaves only the thumb to rove and the pinky is in never never land with nowhere to fret in most cases. This is taught by Joe Collins and Jeff Furman along with many other teachers.
The issue of how best to barre chords on a dulcimer comes up often, and in the end, I think either of the two methods you mention work equally well. My over-riding concern is what comes next and what's the easiest way to get there.I recently had (separate) conversations about this topic with two of my favorite dulcimer players (Linda Brockinton and Nina Zanetti). Nina tends to use her pinky, Linda tends not to. Both play wonderfully and each will admit that there isn't only one tool for job and that there are limitations and strengths with each method
I tend to barre with my index, middle and ring finger and can't recall an instance where this fingering was an issue. While it may seem intuitively obvious that barring with the middle, ring and little finger is "better" because you've got two more fingers to fret with if needed (thumb and pinky), I have found that it doesn't really matter. This is because you *still* have enough options with index-middle-ring (you are not left with only the thumb to rove): you can still use your thumb to fret on any string, yes, but you can also still use your index finger to fret along the bass string (which many people seem to forget). As for the ring finger, in a situation where your index finger and thumb weren't enough and you need to fret somewhere else on the melody string *while also fretting somewhere else on the bass and middle strings using your index and middle fingers*, then I've found the best move in that case it to play out of a new chord position (ie, no longer keep that barre).
Regarding chord inversions, I think they are also very easy to play out of the index-middle-ring barre, either by 1) rotating around the middle finger on the middle string (so for example, out of a barred 3rd fret, play a 4-3-2 chord index-middle-ring, and then play the 2-3-4 chord ring-middle-index) or by using the ring and thumb (Linda Brockinton's method, which can be more comfortable and may better set up for what's coming next in the tune - here you would take that 4-3-2 chord index-middle-ring) and play the 2-3-4 ring-middle-thumb).
So...long winded way of stating that either of those methods work, and work well. In fact, many players sometimes switch between them.