I think you're addressing the symptom, not the problem.
Contrary to your statement, modern washing machines discharge less (far, far less) water than older washers. If the new LG washer is causing the drain to overflow, its either because the drain is corroding on the inside and needs to be replaced, or because the person using the washer is putting too much detergent into it. Modern HE washers need a tablespoon or so of detergent. If you use the same amount that you used to use in an old top loader, you'll produce waaaay too many suds, which will choke up the drainpipe.
Really, I can't stress this enough, use just a little bitty bit of detergent with any modern HE machine.
As for your proposed solution. I have no idea what you're talking about. You say that you already put a larger pipe in, but that the wall won't let you put a larger pipe in, but you're going to put a larger pipe in. ???
How old is your house. What kind of pipe serves the washer?