Is it better to increase speed in a climb?
We know in a climb there is a vertical component opposite lift.
  
  Does this not in effect make the plane "heavier"?
  
  Does this not allow us to keep optimal AOA, and steady state (not
  curving up or down) flight, at a slightly higher airspeed when
  climbing?


No, a plane is in effect "lighter" in a climb than in level flight.  

Climbing "unloads" the wing-- the lift vector is smaller in a steady climb than in level flight

For any given angle-of-attack, the airspeed must be lower in a steady climb than in level flight.

And for any given airspeed, the angle-of-attack must be lower in a steady climb than in level flight.

In a sustained vertical climb, lift must be zero, with thrust supporting the entire aircraft weight (and also overcoming drag). 

Covered in detail (including vector diagrams) here-- Does lift equal weight in a climb?





Note: the question stated that the goal was to maximize "aerodynamic efficiency".  This would seem to imply that the goal is minimize fuel burn per altitude gain, or something similar.  Assuming that the engine is equally efficient at all airspeeds, it would seem that this would be achieved by maximizing the L/D ratio, in which case the concepts stated above are relevant.  Since angle-of-attack is tied to L/D ratio, any shift in the correlation between airspeed and angle-of-attack will change the airspeed that we should fly at, although the shift may be too small to notice in actual practice.  For shallow climb angles the Lift vector is really only reduced by a very small amount, and since Lift for a given angle-of-attack varies according to airspeed squared, the resulting shift in the airspeed associated with a given angle-of-attack will be quite small.

While it's good to understand the basic physics at play, we should be clear that in actual practice, it is by no means always optimum to climb at an airspeed that is lower than the airspeed corresponding to the attack that yields the maximum L/D ratio in level flight.  It depends on what parameter we are trying to optimize and on what other variables are at play.  In most aircraft, these other variables will produce changes in the optimum airspeed for climbing that completely dwarf the slight shift in the relationship between angle-of-attack and airspeed that is caused by the "unloading" of the wing during the climb.

We are more often interested in climbing at the maximum possible rate, or the maximum possible angle, than we are in climbing in the most aerodynamically efficient manner.  Particularly in aircraft with piston engines, the difference can be enormous.

In theory, for a fixed amount of Thrust, the steepest climb will be obtained at the angle-of-attack that gives the highest possible L/D ratio, but this will not be the fastest climb.  Climb rate depends on the difference between power-available and power-required.  Even though the speed where minimum power is required is actually slightly below the best L/D speed, in an aircraft with a fixed-pitch prop the speed for best climb rate is usually above the best L/D speed, because of the greater power available when the engine can turn at a higher RPM.  On the other hand, since Thrust equals power divided by speed, in a piston-engine aircraft more Thrust will typically be available at low airspeeds than at high airspeeds, despite the loss of RPM.  Since climb angle depends on the difference between Thrust and Drag, in a piston-engine aircraft the steepest climb angle typically occurs at an airspeed a little below the airspeed that yields the maximum L/D ratio, i.e. at an airspeed a little below the airspeed that yields the minimum amount of Drag.  For general aviation light airplanes with piston engines, these kind of effects related to available Thrust or Power will usually have an effect on the airspeed that yields the maximum climb rate, as well as the airspeed that yields the maximum climb angle, that completely dwarfs the effect due to the slight "unloading" of the wing by the climb angle, and the resulting slight shift in the airspeed that is associated with any given angle-of-attack.

Example-- from this Cessna 152 data sheet--

Best climb angle 55 KIAS

Best glide speed 60 KIAS 

Best L/D speed-- not stated

(Theoretical best L/D speed is slightly faster than best glide speed because the drag from the windmilling or stopped propeller biases the best glide speed to be slower than the best L/D speed.)

Best climb rate 67 KIAS





Related ASE questions on the relationship between Vx, V best-glide, and Vy:

How can best glide speed be lower than best rate of climb speed??

Is Vy closer to Vbg with a variable pitch prop