I don't think there will be any problems with leaving the plastic mat in place. Since you are not attaching the floor to the substrate, you should have a step sound mat or lumtpapp under the basecogolvet and leave a gap against the walls. Since that type of floorboard is rarely perfectly straight, you should expect that you will need to wedge the boards together to avoid uneven gaps between them. Glue 2 or 3 lengths at a time, wedge the boards together with support against the wall, let the glue dry before continuing, so expect it to take time before the floor is finished.