Also there's no freezer on this fridge
Oops! Being a bar fridge, I really did not know whether it had a freezer or not. Hmm, I normally shot for close to 40 degrees. 45 may be close enough for a bar fridge.
Duh! I just reread your op and see where you said it does not have a freeezer. I can't say I have ever worked on a bar fridge. I guess it just has one stat. And that is a guess. If it is not the stat, and it is fully charged with refrigerant with no blockage in the lines, I am at a loss right now.
Have you checked the light bulb in it? Appliance bulb? Correct wattage? Held the switch while the door is open to make sure the light goes out?
Btw, I usually let one sit overnight when I changed a setting before checking the temp.