Why doesn't a radial burn increase orbital energy? The consensus on the internet seems to be that radial and normal burns don't change the total energy of the orbit, since you're thrusting perpendicular to your motion. I'm having trouble squaring that with the following scenario:

Imagine a satellite orbiting a body at 4m/s. It then performs a radial impulse burn of 3m/s. It's final speed is the 4m/s prograde, plus the 3m/s radial = 5m/s. It's speed has increased, and since the burn was instant, it hasn't changed its position. Thus, its gravitational energy (a function of position) is the same, and its kinetic energy (a function of speed) has increased. Therefore, different orbital energy. 

Where am I going wrong?
 A: I think you are taking some general ideas that hold reasonably well in most situations and finding they don't apply beyond those areas.

The consensus on the internet seems to be that radial and normal burns don't change the total energy of the orbit, since you're thrusting perpendicular to your motion.

If your burn is both "radial" and "perpendicular to your motion", then your motion must be circular.  So a radial burn in a circular orbit does no work.  As long as the burn is small compared to the existing velocity of the satellite, the orbit doesn't change much and we can consider it to still be circular.  You can burn this way constantly and it won't change the KE.  
However if you burn a lot at once, your orbit will change and it will no longer be circular.  Radial burns are no longer perpendicular to motion.  In such a situation, radial burns will now do work.  You can't wave this away just by declaring the burn to be impulsive.  
A: I think for that consensus you found you should always consider a burn with finite thrust. So each applied change in velocity will take some positive amount of time to achieve, during which the attitude of the craft will continuously be adjusted, such that the thrust always points perpendicular to the current velocity. This basically means that you are only changing the direction the velocity is pointing and not its magnitude (this could still happen during a burn, but happens due to gravity).
A: If you increase the speed of the object, or otherwise change its velocity, its trajectory will change, so its orbit will change, and its kinetic energy will increase.

The consensus on the internet seems to be that radial and normal burns don't change the total energy of the orbit, since you're thrusting perpendicular to your motion.

That's hocus-pocus.
A: What a radial thrust will not increase is the kinetic energy of the satellite. The work-energy theorem $\delta K~=~\delta W$ $=~\vec F\cdot d\vec r$ does indicate that if the displacement and force are perpendicular there is no work. Clearly since $d\vec r~=~\vec vdt$ a radial thrust will not increase the kinetic energy.
A radial thrust will increase the potential energy. This will mean the total energy will increase, even though the velocity of the satellite $v~=~\sqrt{GM/r}$ actually decreases with increased radius.
