Is it possible to maintain horizontal flight with the angle of attack above critical value by increasing speed?
That is kind of what the mechanism of lift of a flat plate does: the sharp leading edge destroys the upper flow pattern almost immediately. This question has more info on it.

The lower side of the wing does provide as much or more lift, even when stalled, at AoA of 45°, so lift would not be the problem. Drag would be: 

Image source

On this scale, drag for the NACA 0012 profile is very close to zero until the aerofoil stalls, after which it jumps up and continues climbing. At AoA of 45° the C$_D$ is 1.1 instead of about 0.02 when not stalled - you need over 50 times the thrust to maintain the stalled condition you are describing!