Michael Koblic wrote:
I tried to drill a 3/16" cross-hole in a small steel cylinder (0.3125"
diameter,...

Ummm ... that leaves only a 1/16" "wall" to resist the clamping force.
Maybe the part is collapsing a touch & binding on the bit. Just a WAG.

Try this: clamp with the axis parallel to the vise jaws. 90 degrees to
what you're doing.

Bob