Doors and Windows - window casement repair

My parent's house has double hung, true divided light windows. My mother would like to maintain the integrity of the interior wood trim--there is a split in the wood covering the part of the window casing that holds the lead weights. My idea to save this piece of trim is to back the trim with MDF board (cut shy of the edge of the trim by about 1/2 inch), and glue the trim piece to the MDF with construction adhesive. This would stabalize the piece (in theory), but would it constrict the trim piece too much, and not allow it to "breathe"? The window is over a radiator (in the midwest), and so subject to some wild temperature fluctuations. I'd appreciate any input folks might have, or perhaps a better idea. Many thanks.

LETS RECAP TO SEE IF I HAVE VISUALIZED YOUR WINDOW CORRECTLY...YOU HAVE THE TYPE OF DOUBLE-HUNG WHERE THE WEIGHTS(HOPEFULLY IRON/NOT LEAD)ARE ACCESSED BY REMOVING A WIDE CASING TRIM INSTEAD OF A TRAP DOOR IN THE JAMB,UNDER THE PARTING BEAD. THIS CASING BRIDGES THE WEIGHT POCKET AND HAS SPLIT IN THE MIDDLE OF THIS OPEN AREA?
YOUR SOLUTION WOULD WORK AS LONG AS THE BACKING MATERIAL DIDN'T INTERFERE WITH THE COUNTERWEIGHTS. FOR BEST RESULTS USE ANOTHER PIECE WOOD INSTEAD OF MDF-ALTERNATE THE GRAIN FOR STABLE EXPANSION. WHEN REINSTALLING, NAIL ONLY THE JAMB SIDE AND CAULK THE TRIMMER/STUD SIDE.