Well it's definitely the blue LCD or it's polarizer. Looks like one of them got torched. I've never done this myself, but I have heard some people get a new LCD assembly or replace the polarizer with some of that spotlight gels....either one of those would require some in-depth research online to see if there's any part numbers to help you with.
Sony wants us to replace the engine, and they don't provide part numbers for the LCD panel or the polarizers....but the fact of the matter is, if you have that type of damage, and you want to at least see if you can "Frankenstein the unit" it might just resurrect the engine.
But if we go that route, I'll have to dig into it tomorrow morning. More than likely we'll have to pull the engine out, get the top metal cover off of it to expose the LCD panels, maybe get some pipe cleaners from the Art supply stores, bend it into a U shape and start with removing any dust in between the polarizer and the LCD. I used to clean them that way, and that's a quick and easy job compared to finding an LCD panel or doing a polarizer.
Those are essentially the options we can use to resurrect the engine. I would honestly prefer to send it to a rebuilder and let them do the repair, because they are set up with the tools needed to do it right. $300 for a rebuild....that's fair to me.
I know Sony gave you the options, choose that one, but expect it to be about 6 weeks turnaround time. It could be less, but I prefer to give you the worst, not the best-case scenario.
Now, if you do need more information, I'm here to assist with disassembly. There is a Reply button in this Answer box, use it to get back with me. But for now, I'm going to activate the Answer button, which lets you rate my answer. If you're satisfied with the answer please select one of the happy faces. If not, use the reply button, we'll continue our conversation tomorrow when I can do the needed research and see what is available.