If the presence of a bow quiver is affecting your arrow flight, think about simple things first. Since most quivers mount on one side of the bow, it's like hanging a bit of mass out to one side like an outrigger on a canoe or kayak. If your shooting form has a tendency to 'torque' or twist the bow when you shoot, this extra mass with amplify the twisting motion. So you may see little evidence of torque when you shoot bare bow, but it becomes much more noticeable with the quiver attached.
Even if you have little or no torque in your shooting form, the weight of a quiver hanging out to the side will introduce torque. The torque will almost always be in the direction away from your bow arm. Imagine standing on a ladder looking down at someone shooting a bow. For a right handed shooter the torque, going away from the arm will be counterclockwise pushing the arrow to the left. This will give the appearance of the arrow being too stiff. For a lefty, the torque will be clockwise, and yes, for a lefty this means the arrow will also be too stiff.
I don't think this apparent stiffness has anything to do with the presence of the extra weight added by the quiver and the impact on energy transfer. Extra weight comes into play after the arrow leaves the string. The extra weight helps absorb the energy left in the bow limbs. So the bow is usually quieter and feels more solid and smooth with the added weight.
Bottom line is the arrow will be shooting stiffer with the quiver attached than without.
It won't take much of change to bring the stiffness down. You can add weight to the front of the arrow, use longer arrows, or increase the brace height of the bow to may the arrow fly 'less stiff'. If your arrows are already cut and you don't have easy access to heavier heads, play with brace height. With spine changes you are trying to correct side to side movement. Sometimes changing the brace height can impact the height of your nocking point so as you see the side to side movement disappear a little up and down movement may be seen. So you may have to play with the nock height. It's an interative process with any bow.
Key is to tune your bow and arrows as you will use them. The 'tune' for a barebow will not be exactly the same as one loaded with a quiver full of arrows.