The problem is not with your router unless you have a hole in the roof of your house and the rain is landing on it!
If the drop outs only happen when it rains tere is obviously a problem with our line. You need to re-contact BT and report a telephone fault, not a broadband fault. Tell them there is noise on the line and that it gets worse when it rains.
Once the line is fixed so will your broadband.
You need to be insistent.