The liverpool bit will be fine to ride in, although it might look a bit unusual.
Also if you don't use the shanks on the bit they might get caught on things while riding, expecially trail riding.
I would suggest getting a different bit for riding because, don't you have a different bridle? I would think it would be a major pain to switch bits all the time.
This place has some good, reasonably priced driving and riding bits. www.smuckersharness.com