Here are my thoughts on trying to do both at the same time.
I think it also depends on what kind of Western riding you are planning to do.
For instance, in “English riding,” people can use very different bits between lower level dressage, upper level dressage, hunters, cross country, show jumping, and fox hunting. The horse moves differently, does a different job, and carries himself somewhat differently.
Likewise in “Western riding” there are disciplines like western pleasure and ranch rail classes (one handed, shanked bit, draped rein), there are trails classes (and straight out trail riding), and there is reining, and there is working cows (two handed snaffle).
However, for both dressage and Western a well trained horse is cued for most of the turns and downward transitions from rider body, seat, thigh. That actually translates very well between disciplines, if your dressage horse turns off your thigh pressure or your jumper turns in the direction you look in the arena, then the western neckrein is just confirmation of the cue, not the basic cue.
My dilemma both as a kid and now as an adult, is how do you ride a horse both Western and English at the same time? I totally understand what the posters up thread mean about getting the Western horse totally soft and reactive to the snaffle, but if you train your horse to do this, you may lose the reach to contact and the steady acceptance of the snaffle that is so important in dressage. And if the OP has a 3rd level dressage horse, OP probably doesn’t want to lose all of that training either.
As a kid, I didn’t have consistent dressage-quality contact with the snaffle, though I tried 
and I used a mechanical hackamore for Western, and got the pony light for sliding stops and rollbacks. But for our local playday schooling shows, rules required a bit, so I put on a shanked curb. Western pleasure was riding one-handed in a shanked bit on a draped rein. Big classes and we always lost out to the nice QH, but once, just once, we got 3rd in Western pleasure, so I figure now we must have been within the parameters :).
As an adult I’ve got a big stock horse mare whose basic training is in dressage, and who took forever to accept contact and not go above the snaffle. She is however much happier in a bitless bridle on a loose rein, so I have been using the old mechanical hackamore on her for trail riding (with a jump or dressage saddle). The problem is, there is no direct rein on that bit (she has a nice neck rein and turns off my thigh anyhow) so we can’t practice lateral work very well.
Last month I decided I really wanted to try maresy in cow penning because why not? So I went and watched, then brought my horse to watch, and have now ridden in two lesson sessions.
I got to watch some pros warming up for the Calgary Stampede qualifiers, and I totally see what the poster above means by getting the horse super light off the snaffle. They were doing a kind of bump bump upwards two handed with high hands, and indeed the horses were also super responsive to leg and weight, so that I saw one rider practising patterns with the reins dropped on the horse’s neck. They also need a snaffle to point the horse’s head at the cow, and they ride two handed.
I went in with the mechanical hackamore. Maresy didn’t go beyond a trot the first night, the second session she really got her head into the game and started cantering, and that’s where the mechanical hackamore doesn’t quite allow for fast stops and turns.
But I don’t want to use a snaffle on her, I want to save that for dressage. Anyhow, the solution I am going to try next session is something my coach found for me, that she calls an “English hackamore.” It’s just a mechancial hackamore, but it has shorter shanks and doesn’t have the stability chain on the shanks, and has a softer nosepiece, so it absolutely works as a direct rein and you can ride two handed. I tried it out yesterday and we can do lateral flexions and shoulder in with it.
Yes, I think putting a small Western snaffle with a chin strap on maresy and doing bump bump would be more effective, and I also think she would appreciate the invitation to go above and behind the bit. But that would finish us forever for her dressage foundations. Also I think she might get a bit pissy about being bumped in the mouth at speed. And the cattle penning is something we are doing “off the side of our desk,” so to speak, without much investment other than to have some fun.
Anyhow, the bit you choose for Western riding is going to depend on what Western discipline you choose, and how much “performance” it involves. I think a 3rd level dressage horse could do a Western rail class in a shanked bit on a draped rein just fine, especially if the horse has had a double bridle on before. If you wanted to be really competitive, you’d need to get the horse to slow down a bit and do a stretchy trot.
But I think if you want to do reining or cow work, you will need to train in a two handed bit with direct rein.
