you will overdraw your bow with a high BH, which may not be made or tillered for a certain length and BH.
I don't think that's true at all. Your draw length doesn't change, but your EFFECTIVE draw will change. Read Tim Baker on Bow Design and Performance. If you have a 26 inch draw, and brace the bow 4 inches (very low, i think), your Effective draw will be 22 ". If you brace it 6", then obviously you are only pulling it 20 inches to reach your draw length.
If i'm not mistaken, this means that more or less units of muscular energy will be imparted to the limbs of the bow. The energy required to hold a bow AT brace height is not part of the equation. The proof of this is easy: nock an arrow, but don't draw the bow: when you let go the arrow either falls to the ground (loose nock) or stays clipped to the string (snug nock). This, I guess, is why a bow braced at lower heights has better cast: more muscle energy is available to propel the arrow. But, as Pat says, the lower the brace height , the more you slap your wrist.