Indeed. It's a reasonable short treck, and unless you're doing it in the middle of one of the arriving "waves" when some aircrafts land at E at near the same time, you're not expecting any queues to do so.
I'd say leave the E Gate Lounge 45min before departure will be plenty.
Going via the unclean (but no passport checks) way could be even slightly faster (but well, no one likes security checks) but we're talking about differences of minutes, if not seconds.
When your flight departs 5pm, leave the lounge at 4.15pm and you should be good (roughly when it displays "Go to Gate" or a bit earlier)
That said, 1hr50min will give you "just enough" time in the E lounge. When you arrive clean in A, it will take you about 10-15min to the E lounge, so timely arrival expecting, you should have about 45min to spend there before heading out again. Just enough (plus burning those extra calories will allow you some more food
) time I guess, and you don't seem to mind the trek for it, so I say go for it.