No idea what alternates are allowed, or what ETOPS distance you can use-- but if you can use 800 nm from St Johns and Bermuda, then...
World Aeronautical Database says CYTT is at 47.618610N, 52.751945W and LPLA is at 38.761842N, 27.09079W. It turns out that 36.16543966N, 43.54313521W is the place that's 800 nm from both of those positions. It turns out that 34.58778995N 48.94366471W is 800 nm from St Johns and Bermuda, assuming the latter is at 32.364042N 64.678703W. So you need to stay north of those two points, and at a glance it looks like your route does so. You can confirm that by using the NGS calculator
INVERSE computation
to find your initial course from 37N 40W to 35N 50W is 259.15 degrees, while the initial course from 37N 40W to the first corner of the no-fly zone is 254.78 degrees and to the second corner is 254.34 degrees. So you are indeed passing north of both of them.
Once past the second corner you're always well within 800 nm of someplace.