Changing strings is not difficult at all.. It just takes time. If you take them all off at once then the bridges wont stay put as all that keeps them in position is the strings. Thast no big deal eaither if you take the time to either mark the position of them with tape or measure from some spot. After restringing you will have to mess aroud some with that treb bridge to get it exactly right. Again no big deal other than taking time.
If the bridge rods/caps and/or side rails are in bad shape, replace them also if you do all the strings at once. It would also be a good time to refinish/clean the dulcimer if needed too.
If all you need to do is to replace strings, then do it a few at a time. That way the bridges stay put and you dont have to go through a day or so of dulcimer withdrawel.