I just installed a GE 8300 BTU thru-the-wall unit in our master bedroom. Used the GE shell that is specified for the unit.
We like it alot. The only qualifier is that the GE unit is more noisy than a comparable LG unit. So, if I were doing the job again. I would use the LG unit and its recommended shell. (About $200 less than the GE unit.)
Also: If you cut a hole thru a load bearing wall, whether masonry or lumber, you MUST add structure. (Lintel), over the opening. It sounds like you are going to punch a hole thru a wall that is about 12" thick. Don't try to reinvent the wheel. Get one of the thru the wall AC units.