Yep, the French braid should be fine. Just braid it tight up to the horse's crest, keep it tidy. The forelock should be braided as well. Sort of french braid it down the area where the hair grows from, and finish down the length of forelock, tie it off, then fold the braid under itself and tie off again.
Be aware, though, that Frenched manes pull loose if the horse puts his head down. I braid the forelock and tail the night before, my horse spends the night in leg and tail bandages and a lycra hood, and I braid his mane at the show grounds. After the mane is braided, no grazing, eat only from a hay bag.
I do like to french braid the tail as well. It just completes the picture, and sometimes you can gain a few points in the class for putting in that time when other entries didn't. The "good" tail picture that Anebel posted is what you're shooting for, but instead of the pinwheel you can
simply finish the french braid and tuck the "tail" of the braid neatly inside the braiding over the tail bone. I can't keep pinwheels in and looking good all day
I see a ton of French braids on English horses in the ring, even at some of what we consider in my area to be bigger open shows. I French braid my quarter pony to avoid cutting his mane, he lives a partially outdoor life, and needs the hair.