Combining tile layers in different projections is problematic
I haven't tried, but I think this it is not possible to combine tile layers in different projections.
Assume that your map is shown in EPSG:3857 and you have one tile layer in EPSG:3857 as a base layer. You now want to add another layer of images on top (that's what a tile layer basically is), which comes in a different projection. Every pixel of these images should be positioned correctly relative to the base layer.
To achieve that, you would need to re-project the images, so that the individual pixels get positioned correctly. Consider e.g. the different distortion of EPSG:3857 and EPSG:4326 at high latitudes. So you cannot deliver pre-made images, but need to change them so that they fit the underlying projection. That is simply not what a tile layer is meant for.
You will need to adjust your workflow so that images are served in the projection they will be shown in.
This feature request is another hint that it is not possible: https://github.com/Leaflet/Leaflet/issues/1450
Some more updates
The main problem is that a square in plate carrée, i.e. a tile in EPSG:4326, does not cover a square area in "Web Mercator" EPSG:3857. Even if you position e.g. one or two of the corners correctly, most of it will be wrong relative to the underlying EPSG:3857 tiles. The only exception is tiles very close to the equator, where both plate carrée and "Web Mercator" squares actually closely resemble squares on the earth surface.
Note that there is a big difference to positioning a vector layer in another projection on top of a raster layer: For vectors, each point in the geometry will be re-projected separately, which is what you would need to do for your raster layer, too. There are also cases where two projections are related to each other by e.g. just a lateral offset. In these cases, a simple translation (shifting) of all tiles in one layer would be sufficient.
The proper way to solve your problem is to re-project your imagery to the CRS it will be viewed in. Apparently you have FME, which is a great tool to do this. This question is not about using FME, but to me it seems as if you do something wrong in FME if your results are so bad.