I see no problem with riding with side reins as long as you know how to use them properly and some else is there with you. My instructor had me put it on my OTTB when I was training him to ASSIST in getting him to bend. There was no contact from them only the weight and its up to you to ask for the bend.
As for the comment on how harness horses have to carry their heads high? Thats not true. Some are raced without checks. Checks are used to help with controlling the horse, some really get ridiculous in a race. Yes some people abuse the check but it is also a balancing mechanism to assist the horse.
If all else fails, get a private instructor out a couple times to see what would work best for you and your horse. I wish you luck!
Also if you at all think it has something to do with back pains, have a vet come out and do a quick once over to make sure everything is dandy.
A Dog looks up to a man,
A cat looks down on a man,
But a patient horse looks a man in the eye and sees him as an equal