Yeah, a saddle fitter, I know it's a pain but if you widen the saddle too much it will drop forward. It could be causing him pain, plus a breast plate will do nothing to keep the saddle in place.........I tried that one too ....
A mare I know had a tipping saddle, physio out and she's protecting a foreleg causing one hip to drop. This is caused by a foot balance issue. It's worth getting the horse looked at!
If the saddle slips there is usually something wrong with either the rider or the horse IME. Not that it can't often be fixed, but maybe one or both of you needs to see a physio. I would have the saddle fitter out first though, as it is possible that it has become damaged, but after that, if the saddle tree and stitching is sound, then I would then have the physio.
I have found that you may well find that the stuffing has compressed more on one side, but this will be usually as a result of the saddle starting to slip and so having more weight on one side, rather than the primary cause. I have known saddlers "stuff up" the compressed side, but this does not usually on its own stop the saddle slip, just make it less noticeable for a while as that side is stuffed up!
Once you have a diagnosis on what is causing the problem then re-assess and maybe have the saddler put a balancing girth strap to stop the slip. Think like a side saddle but less radical. I would not however do this other than once you know why the saddle is slipping in the first place so you can start work from the cause up.
You may find your trainer can help, or I use a trainer who is also a Cranio Sacral worker, and she can see the issue and put me right before the saddle starts to slip, so it is still dead square. If you wanted to look deeper look up Russell Guire, he has top diagnostic equipment.
One thing I would not do is just stuff the saddle up, as you and the horse can over time make each other one sided! Also there may be a start of a more serious problem that could be fixed if you flag it up before the horse is "lame".