Its really rough along both sides. Do you think by cutting and replacing the bottom on both sides would work as seen in that pic? I think that would be easier.
I've done both. Cut out bad wood, added in new wood and used bondo to fix a broken corner. Both methods work, but you need some sort of rebar in both cases. I use 1/4 oak dowels when adding new wood, and you can use screws driven into the edge of the cabinet wood for the bondo.
With the dowels, you wanna first cut out the frayed wood and cut a straight edge into the existing cab wood. Then cut an
oversized piece of 1/2 plywood, favoring the outside edge if the new material is slightly thinner than the existing cabinet wood. Mate it to the cut edge on the cab and use bar clamps to hold it in place. Now get out your hand drill and a 1/4" inch bit and drill 2-3 deep evenly spaced 6-8 inch holes edge-on thru the replacement wood and into the cabinet side, making damned sure to keep your drill bit centered and parallel to the outside and inside planes so you don't break thru one side or the other. Now, remove the clamps, add glue to the holes, insert the dowels. Now add some glue to the holes in the new piece, then add a bead of glue to the cab edge and the new edge, and slide the new piece onto the dowels. Now reclamp the edges of the new piece, using the pressure to drive the new wood into the cab edge, using the bar part of the clamp to keep the 2 pieces plumb and in the same plane. After the glue cures, use a straight edge to mark the edge lines and use a router with a flush trim bit to trim the excess wood to match the original cabinet edge, following up with a sanding of the edge.
With bondo/screws, you wanna clean out the loose wood first. Then drive the screws, (usually 3-5 " deck screws) into the cab edge about 2 -3 inches, leaving enough sticking out to strengthen the bondo. Make sure to drive them far enough in so that they don't stick out into the area near the new edge where the t-molding slot needs to be recut, as router bits don't like screw heads. Now you'll need to make a form out of plexi or plastic or some material that doesn't stick to bondo. Now build the bondo up in 3-4 layers. when you are satisfied, remove the form, fill in the remaining voids and sand flat, using a straight edge to get the edges true.