AppleCare about to expire - Best plan of action?

I have a mid 2012 MacBook Pro, currently running Yosemite OSX, 10.10.5.

As with many of these models, I have the dreaded screen glitching, rebooting, freezing, three-beeping problem but currently it is intermittent (1-2 per week), occurring only when I either press firmly on the bottom of the computer (e.g. when carrying it across the room holding the left hand side of the machine) or apparently randomly.

Reseating the RAM or even just undoing the bottom cover and tightening the RAM bank screws seem to fix it temporarily. I have upgraded the RAM and placed a new 500GB SSD, both from Crucial, in the machine, so I know it is not a RAM problem itself.

This problem appears widely reported in this model of MBP, e.g. see here: Picture Included. Mid 2012 13 inch Macbook Pro green/purple lines on screen. Freezes. Sometimes 3 beeps. See on the second page, one reviewer has provided a meta-bank of similar posts and has urged people to send Apple feedback about this problem (which I have).

I currently have AppleCare warranty for another month, so could just get the logic board replaced. However, there appears overwhelming anecdotal evidence that a replaced logic board does not fix the problem - it reoccurs at some point afterwards. With only a month left of AppleCare, I want to make my case that I am entitled to a new machine as i) this problem is over-represented in this model, ii) I have taken exceptional care of this machine and upgraded it with new parts, and iii) there is evidence that a new logic board does not fix the problem.

What are people's suggestions? Should I just go for the new logic board and hope it lasts another 1-2 years, and then try and deal with it myself thereafter (i.e. replace logic board myself), or is it possible to make a case to Apple that I should receive a new machine?

Thanks in advance.

Message was edited by: cmg212