NSolve does the job by
NSolve[E^(4 - 2 x - 2 E^(2 - 2 x) x) x - x == 0 && Abs[x] <= 10, x, Complexes];Dimensions[%]
{279, 1}
Addition. It's strange that
NSolve[E^(4 - 2 x - 2 E^(2 - 2 x) x) x - x == 0 && Abs[x] <= 10, x, Complexes, WorkingPrecision -> 30];Dimesions[%]
{268, 1}
and
NSolve[E^(4 - 2 x - 2 E^(2 - 2 x) x) x - x == 0 && Re[x] <= 10 &&
Re[x] >= -10 && Im[x] <= 10 && Im[x] >= -10, x, Complexes,
WorkingPrecision -> 30];Dimensions[%]
{268, 1}
I'll try to verify it with Maple.
Addition 2. The verification
Table[E^(4 - 2 x - 2 E^(2 - 2 x) x) x - x /. sol[[j]], {j, 1, 268}]
confirms all 268 solutions.