I doubt its the GPU, thats not as common as the internet likes to think it is. If you have another Mac, I'd recommend installing MacOS on a USB drive and see if you can boot off of that. From there, hopefully you can use the external monitor to recover your data. Then you could create a MacOS installer USB to try to reformat the SSD and reinstall MacOS. This probably won't solve your display issue though, only the failure to boot that you're seeing with the external monitor.
You may have two faults, the first being the LCD which is why Apple said to get it replaced, but possibly also the operating system may have become corrupted. If you've been turning it off and on over and over while its trying to boot, you may have accidentally corrupted it while trying to figure out the no display issue (computers don't like to be interrupted while trying to boot).
Unfortunately, this is probably a good situation to either bring it to the Apple Store or a local repair shop who has parts on hand to test the various possibilities.