A little wiggle when the baseplate is off is normal. The spring loaded pin can become a bit sticky. You may be able to pull the pin across to apply more pressure to the receiving hole, but as long as the back door is not falling off, there is no issue.
Do not fill the holes or add washers. The back door needs to move so it can centre itself when the base plate is installed.
The back door should be firmly held in place with the base plate on.
You have Kanto Camera in Kawasaki. I would take it to them if you are still concerned.