What materials can be used to prevent polyurethane glue from sticking to unwanter surfaces?
The following is a carry over from a CNC video I saw many moons ago, but have lost the link.

Give the selected adhesive a sacrificial material on which to bond. Your thoughts of using plastic bags aren't far off, but let's add in a layer of adhesive that you can remove.

If you cover the clamping boards with wide painters tape where the glue is going to contact them, you'd be able to remove the tape from the clamping boards leaving the adhesive side exposed and likely mostly shredded.

If it's practical and possible to apply the tape to the shutter boards with the adhesive side pointing at the shutter boards, you may also reduce the amount of cleanup, as the tape will peel off, again possibly in shreds, leaving behind the tape adhesive and the extruded glue from the shutter joins.

You may be able to minimize the extruded glue by taping on one side the shutter boards along the entire length of the join while held together, then bending the taped joint open and applying the glue. When the boards are flattened, it would be easier to wipe off the glue from only one side, then apply more tape to keep that side from bonding to the clamping boards.