Keith wrote:
"Connect a length of 50 Ohm transmission line to a 9 Volt battery and
wait for the transient to die."

The transient is the only part of your proposal for which the 50-ohm
surge impedance has meaning. That`s why it`s called the "surge"
impedance.

The 50 ohms is Zo and equals the sq.rt. of Z/Y, or in a perfect line
this is sq. rt. of L/C.

L and C require an alternatng current (changing, not static) to produce
reactance, so the d-c calculations are meaningless to Zo.

Best regards, Richard Hsarrison, KB5WZI