I think you mean deflector, not chute. Every mower deck that has a side discharge has a chute that is the opening to allow the clipping debris out from under the deck. The chute is an integral part of the deck, but the deflector is typically hinged and allowed to rotate up and down.
There are a couple of missing pieces of information: (1) Kind of mower, (2) Kind of deflector.
I have a John Deere 717A, 48" 7 Iron mowing deck (a ZTR). The deflector is an important part of the mower deck. The clippings are distributed much better with the deflector down, or rained an inch or two at the end. Mowing conditions dictate the best position of the deflector. The worst position is the deflector to be in the fully upright position (e.g. no deflector). With no deflector, a windrow of clippings is left 4-6 ft away from the discharge chute.
I have a small chain with a hook mounted from the frame to the tip of the deflector. When transporting, I simply pull up the deflector, use the hook to maintain a fully upright position. This means the smallest width possible, and allows me to put another mower beside my ZTR. Keeping the deflector down means wasted trailer space.
On the other hand, I have an Exmark w/b, TriVantage deck. The deflector is a useless part of the deck. I have it mounted with a loose pin, so that it is easily taken on/off, or at least swung fully up. Only rarely do I operate with the deflector down (e.g. when I don't want clippings to be spread wide). But, it leaves a windrow with it down.
You need to make your own decision on the value of the deflector. You may need to run with it in tight spaces to minimize the chances for debris flying in unwanted directions. Or, you may need it for best clipping distribution. If you need to keep it, try to make a provision for locking it in the upright position to make your mower the smallest width. This may eliminate your concern for trailer constraints.