The insulator is right and knows his stuff. You are dealing with 2 different barriers. Moisture barrier on the outside wall and a vapor barrier on the insulation. Years ago HUD came out with a requirement that HUD homes had to have a moisture barrier on the basement wall before putting on the finish wall. Unfortunately, different climates have different problems. 2 barriers together will trap moisture as you have found out. I think you need to address the worst problem and then remove the other one.
If you have water coming through the walls, then the moisture barrier may be more important. However, there are other ways to fix this including sealing cracks that leak and spraying a moisture proof coating on the wall such as UGL waterlok. A moisture test you can perform on the walls and floor is to tape a 2x2' or 3x3' piece of plastic or tin foil on the wall for a couple days. Seal it all the way around with duct tape and later look to see if moisture is underneath and how much. Many people have humid basements but I probably wouldn't use a moisture barrier unless you usually see moisture on the walls.
You could do a partial removal of the moisture barrier depending upon where the moisture is coming through the outside walls. Good luck.