I use about the most mis-matched tack imaginable, luckily (I suppose) my behind covers most of my purple saddle.
There are very nice western cavessons readily available. I have a nice one with a rolled leather nose, note this is not the same as the ones used with a tie down (don't know what they are called). Used with a western headstall they have the same purpose and effect as an English one.
I have also used an English cavesson with a western headstall.
Horse's don't give a rat's rear end, and neither do I.
I don't ride an English saddle often right now, but when I do, I'm using a Navajo blanket underneath, Aussie-style 2.5 inch leathers with western roper stirrups. Add in a rope sidepull halter, jeans & cowboy boots, and I guess I looked...confused?
But my horses don't mind, and neither do I.
I've never used an English bridle, but if it made my horse happy, I would.