The subdistrict municipality Bueng Kan (เทศบาลตำบลบึงกาฬ) was created in 1956 as a sanitary district [Gazette], and upgraded to a subdistrict municipality in 1999, when all of the sanitary districts were upgraded. It has just a population of 4653, as it covers only parts of the subdistricts Bueng Kan and Wisit. With this small population it'd be by far the smallest town - for normal towns the population has to be at least 50,000, so I was expecting only the provincial capitals to be below this margin - but that'll be a topic for another posting. But Mae Hong Son as the smallest provincial capital is already a very small town, it has just 6525 citizen, so such small towns are not completely unusual.
However, if I read the Thesaban act of 1934 correctly, it states that the municipality which qualifies to be a town is the one in the subdistrict which contains the province hall. But for Bueng Kan, the province hall will be built in Non Sombun subdistrict (though the district office will be the temporary province hall at first), so that would mean the TAO Non Sombun would be the one which qualifies to become a town, not Bueng Kan. On the other hand, there are at least two province where the province hall has been moved outside town recently, i.e. Nakhon Pathom and Kamphaeng Phet, but in these provinces no changes has been made with regard to the municipal status of the area which now houses the province hall. I don't know if there were any changes in this specific item in the Thesaban act in any of the later amendments which would clear up this issue.
With a little help from the Wikimapia map of the area, I have done a map which marks the main office locations of the new province, I only don't have yet the exact site where the province hall will be built.
View Bueng Kan administration in a larger map