i'm not acquainted with the rat king, but for bernard: you can get 100 likes and 3 hearts
In regards to the Rat King; I've done everything I can (Using saves, to enure the best choices) and I've reached a max of 53. The 'Rat Problem' quest is currently incomplete, so I gurantee your relationship with with both the Rat King and Scrap will be improveable, in future versions.
In regards to the Bartender, most of your relations kind of rely on your bitch fame for the town, as well as his jobs. However, at the moment, you can't really get anywhere with him. So, yes, the max is around 35.