I've been a number of times in Mukdahan and can suggest:
1. There is a mechanic I've used several times and he is a descent guy. It's difficult to explain but - if you exit Tesco Lotus, you come to an intersection with a traffic light. He is on the corner opposite lane. He's got a few bikes in various condition sitting outside his place.
2. If you can leave it in Savanakhet, a place to do so is with Louis at Louis Motocross. From Lao-Paris cafe, go upriver to the next Wat, turn right before the wat or after, on the next street up, go left and he is on the right side of this one way street, you'll see a few bikes parked outside and quite possibly his 250 MX bike there. Tell him I sent you.
3. You may be able to leave it at the customs house on your entry, inquire at the bridge. There is a secure lock up there for impounded stuff.
4. You can try another mechanic in Mukdahan I know. Directions: from the main traffic roundabout go past the hospital and follow the road all the way around until you come to the traffic light by the junction for Hwy 212. At the light, go right (direction Tesco) and he is about 300 meters down on the left. I'm not sure he remembers me, most likely not as I only got service there three times a few years ago.
5. The Grand Hotel has an underground parking lot, inquire with them but it is a big draw for the disco crowd on weekends, lots of people and security is focused on troublemakers.
6. Along the second road parallel to the Mekong, drive north and a bit down from the 7-11 is a little guesthouse on the left occupying the corner. It is a drive in courtyard, inquire with them. Sorry, can't remember the name of the place.