Measure the L value by ensuring that, at the far end, the two wires are
connected together (i.e. a short circuit at the far end) and using the
relevant range on the meter.
Should it be half this value to work out the velocity of the cable?

Check the R value using a reistance range, with the wires shorted as for
L.

The results may, of course, depend upon your meter to some extent. So, may
depend upon the frequency it uses, how long the cable is (in wavelengths),
and if/how it takes the series R into account when measuring the L, and so
on.
He wants an LCR bridge. Don't typical handheld LCR meters use a frequency
around 1MHz?

A quick check is to work out the signal velocity from the C and L values.
This should be approaching the speed of light in vacuum. If it exceeds
this, or is much lower, the values should be regarded as unreliable.
What is the velocity factor of typical twin speaker cable? About 0.67?

Pete