The symptoms of a bleeding episode, contrary to popular opinion, do not actually include visible blood! The blood is in the lungs, deep down inside, and can only be seen by scoping withing 30 minutes of the exercise - after that, it will have been reabsorbed by the lungs.
The usual symptom is fatigue and "running out of gas." The horse can't get enough oxygen, so he can't run as fast, so he feels logy and out of gas to the rider.
Another consequence of bleeding is often respiratory problems, like colds and snotty noses.
It's impossible to know whether he's going to be okay for what you want without knowing how badly he bled. This is a progressive thing; once a horse has bled he's probably going to do it again, and it'll get worse.
On the other hand, probably 80% of the ottbs out there have bled, and most go on to athletic second carers without problems or medication.
Do you know what meds he was on? If he was on more than one, that may be a sign he's a worse bleeder than a horse whose bleeding was controlled with only Lasix.
I would go ahead and see what he can do, building up carefully and not overtiring him. If he runs out of gas, don't over encourage him to do more. If he gets a respiratory problem, give him some time off, call in the vet; you'll probably need some antibiotics.
Learn to watch his breathing. Count his respirations per minute and compare to a chart of "normal" respirations. When he's standing in the stall, it should be slow and regular and easy. When he's exercising, it should be faster, but still even and not labored. The recovery period - the amount of time it takes to get from exercised to relaxed - should be reasonably short. You'll quickly learn what's normal for him, and learn when he's overtaxed.
To count the respirations, look at his flank near the stifle, and count the big breaths - if he's eating or something, he'll move his flank in and out for more than breathing, but it should be obvious. THis is easier than trying to count his nostril flares, at least for me! :)
Good Luck with him. Most ottbs will be capable of what you want.
Edit: Just want to add... for the vet to assess his bleeding, he'll have to be exercised for her and then she'll have to scope him immediately after hard exercise if she's to see anything at all. So if you want a vet assessment, let them know in advance so they can tell you what to do.