I agree that it depends on the horse .... somewhat.
The bigger factor is going to be how you TRAIN the horse for both, and how you give the horse cues so it can tell the two apart.
And will you go win the National Finals Rodeo in barrel racing, and also go win AQHA World in Western Pleasure? Heck no. So often the horse won't be a superstar in both, but maybe be good in one, and *okay* in another. So as long as your expectations are realistic.
My old horse Beau in Western Pleasure: (This is an old photo, but I want to say we took 3rd or 4th that day.)
And my old horse Beau in Barrels: (Took 4th place this day at a family rodeo with about 30 barrel racers)
And my current horse Red:
With Red, he's still picking his speed up on the barrels, so I hauled him a lot this summer to local shows. At the last showing show, we took
--3rd place halter
--3rd place western pleasure
--3rd place reining
--2nd place trail
--2nd place pole bending
--1st place ring race
There were about 12 to 16 riders in each class.
For the slow classes, I have a solid low-port bit that I use. And no breastcollar and no sport boots. My body position is heel-hip-ear and sitting tall. I'll give him calm verbal cues of "easy" and "walk" so he knows we're going slow.
For the speed classes, I have a Carol Goosetree Simplicity bit that I use, along with his breastcollar and sport boots. And my body language is much different going into the arena. I also "talk" to my horses before I go in. I'll whisper and ask excitedly "You ready Red? You ready?" And that's kinda his cue that we are going to go in there and go FAST.
I also feel like it has been easier to teach him to go slow BEFORE we are going full blast on the barrel pattern, and teach him many different events at the same time. Rather than if he was already a fully trained barrel horse, it would just take him longer to figure out the WP slow classes.
But absolutely. It can be done.