so you will need to install the 3/4 and then slice another rod in half and cram it next to it. you need to dummy up the hole with filler.
Could you consider using molding instead? Use quarter round or something to dress off the area and also hide the crack.
Could you not "rip" a piece of 1 by to wedge into the crack?
Caulk might be not the best choice with a one inch crack, what about non shrink spackle for interior? What about Rock Hard? Both are non shrinking and will span the area with backer rods.
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