A horse can have the worst conformation ever and be a jam up 1D barrel horse.
Likewise, a horse can have textbook perfect barrel racing conformation and not even hit the 4D.
Now with that said, it is nice to have a prospect that is well put together because that decreases your chances of having lameness (of course, does not eliminate it) but most barrel racers will tell you that the HEART of the horse is what makes a top barrel horse.
Your horse is weaker in the hind end than I like to see in a barrel horse, but looks like she has clean legs. She may have a long-ish back (which makes it harder for them to collect those hind legs under them) but it may also be the strange way she is standing.
And when you say 3D/4D times -- are you talking about local jackpots or large events?
If you are just looking to have some fun with your horse, I don't see any reason why she couldn't do it, so long as the horse enjoys it.