Probate records for Roenne city are only available online up to ca. 1920.
Concerning the mayor issue.
There was no elected mayor “of” Bornholm at that time, but an appointed “Amtmand” (county bailiff) – a high ranking, appointed civil servant. So you can leave out that option.
But was he a mayor “on” Bornholm (somewhere)? Well, he lived in Roenne, and I checked the list of mayors at the online city archive for Roenne - no luck. He may have been a member though of the elected city council and had a position as vice-mayor. To check if he was ever a member of the Roenne city council you’ll have to find a helper living on Bornholm willing to check at the city archives.
Do you have a photo of him? I attach a link to the city archives showing some old pictures of city council members.http://www.roennebyarkiv.com/byens-styre.html