Would retrofitting weep holes reduce dampness?
The answer is pretty much, yes. I think someone with experience needs to see what you are dealing with (the "level" of the moisture). The damp course should be preventing moisture issues... my question would be why isn't it, or why do you have so much moisture, or is this actually a moisture issue? There might be some other issue, like rain leaking into the wall cavity.

Weep holes and/or vents could help (if they are necessary). A brick layer should have the experience to do this. It is not terribly complicated procedure; I would expect many construction workers would have the know-how to perform the "surgery". The one thing critical to having any contractor work on your house is that they are bonded and insured.