Hi,
Ah L&R. You have a bit of a problem to solve but one I commonly encounter on their locks. The primary problem is they do almost no real fitting of their cast parts. There is a raised collar on the lock plate side of the tumbler. You may have to turn it down to bring the tumbler closer to the plate. However, the very first thing to do with any L&R lock is to make sure the inside of the plate is flat. With a flat file close to the width of the plate, file the surface. The first strokes will reveal the depressions.
The smooth and polish it before trying to fit the parts. Here is a typical L&R tumbler:
It does not look much better than some India-made guns and requires a lot of clean up and truing. I spin the tumbler in the chuck mounted on my wood lathe and true the sides with a file that has safed edges. A safed edge is rested on the tumbler post and the bridle spindle and the file flattens and trues the sides of the tumbler. You have to anneal the tumbler first to do that. The link below may help you. It discusses working over an L&R Queen Anne lock but many of the problems are likely to be similar to yours
Reworking an L&R Queen Anne lock FINISHED
Unfortunately, you rarely can do just one thing on a lock without affecting something else so be prepared to really work your lock over.
dave