I would not recommend a fixed sided bit, for driving. It makes the bit less “able to talk” to the horse, in my opinion. If you can move one side, while the other bit side stays in place, you are doing two things with it. Holding the still side of horse in place, while maybe bending the other side. Gives the horse a feel to aid him in holding position.
We call fixed sided bits, ridden or driven, H style, because that is what they look like. In years past, olden-days, these were the most common bits because theconstruction, manufacturing skills needed for the movable sides, were hard to come by. Made them more expensive. Horses were not expected to do the activities we ask of them now with bending in the head, neck, shoulders, barrel, haunches, independently. With the H style bits, if you touch one rein, the whole bit is affected, horse can not get any subtle signals because the whole bit moves at the same time. No flex with straight back pull.
I know some folks still like the fixed side, often pointed at as Multiple bits. I had a runaway once, back in the beginning of my driving career. That H style allowed no ability to turn, pulling one side just made bit go sideways in her mouth. I ended up running her into a tree. We have tried using some of our old bits as a solution now and again, but always end up feeling like horse is not able to understand us, because we cannot finesse the reins with the H style bits.
I think the many options in bits that allow each side to communicate apart from the other bit parts, are vastly improved over the old designs with fixed bit sides. Ridden, my horses are better able to understand what I ask by lifting one rein, one side of the curb bit. Driven they are going to be able to work with me, though perhaps a little limited with no rein out to the side as when ridden. The lines of reins, bit, shoulder rings, saddle terret, hands, can force the Driver to ask differently than the Rider is able to. Horse will be bringing his ridden skills, understanding of the bit, to Driving and usually will “get it” quickly when moving to body bending, flexing and going into collection when asked from the carriage seat.
I consider better bit design one of the major improvements in the horse activities over the last couple decades. Bits that talk better have allowed that to happen.