well most tropical fish range from 60-82 , but again depends on species and how tolerant they can be.
Personal opinion gradually turn up the heat a degree a hour or so and watch how the fish react. If you see anything out of the ordinary that you see now. Then Id turn the temperature down, but I think you should be alright if you slowly increase the temperature to 82-85. Do it to fast you can put the fish into shock and lose them all. So gradually increase temperature and just keep a eye on the fish.
I recently got a case of ick and I turned the temperature up to 85 gradually and all fish are doing well and accounted for.
Barbs in general are pretty tough fish, hatchets aren't as durable as barbs, but that's my opinion.
You should be fine