OK, I found a solution.  I do this in onCreate:

LayoutParams lp = button1.getLayoutParams();
lp.height = dispWidth / 5;
button1.setLayoutParams(lp);
button2.setLayoutParams(lp);
button3.setLayoutParams(lp);
button4.setLayoutParams(lp);
button5.setLayoutParams(lp);

where dispWidth and dispHeight were previously set by

Display display = getWindowManager().getDefaultDisplay();
dispWidth = display.getWidth();
dispHeight = display.getHeight();

also within onCreate.

This solution lets the layout_weight="1" and scaleType="fixXY"
properly scale down the image and divide up the width (which just
happens to be equal to the full dispWidth) and then I change the
layout params to include a calculated dispWidth/5 to be used as the
height, replacing the "50px" default.  This leaves the buttons always
square and always exactly filling up the bottom of the display

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