The only thing that you should think about as far as a conflict between dressage and Cowboy Dressage is the relationship with the bit that you’ll need to exhibit for each way of riding.
The “soft feel” (which has several layers of meaning) is the signature aspect of what Cowboy Dressage was built to promote. It will mean being soft or light on the bit. It does not necessarily mean behind the bit.
What could be hard for some lower-level, dedicated dressage horses and riders is how to resolve the “soft feel” relationship with the bit with the firmer, more consistent contact that many lower-levels horses show if they are (correctly) “through,” really demonstrating power that starts in the hind feet and comes up to the bit, correctly using their top line.
A reasonable way to handle this for a well-trained dressage horse is to allow him to travel a little bit under tempo and, really, with a bit less power such that he isn’t pressed up into the bridle so firmly. See what I mean? You can have a lighter feel in your hand, a nice, open throatlatch and still a good, slightly uphill poster just by asking for only as much power from the hind end as you can contain in that soft feel you have in your hand. But you need to be pretty good with your hand so as to encourage stretch forward into that light feel since you cannot produce it by riding with more power into the bridle.
I am one of those riders that can figure out how to put a well-trained horse’s head just where I want it, and I can create some semblance of the feel in the hand that I’m told to. With a reasonably made horse, I can change up between the Cowboy Dressage and dressage ways of going. But not everyone or every horse will find that so easy. I just thought I’d describe this difference and how you might get it in case it makes sense to you and your horse.
Have fun! I have met some nice folks in the Cowboy Dressage World. And the tests are harder as well as more fun than they look.