The diff assembly of a Healey is on an aluminium carrier, which makes it rather noisier than if it was cast iron. Therefore assembly by professional is preferable.
in the UK I have a friend who takes the crown wheel and pinion to Guest Gears who make them on the machines R-R used for theirs, so probably American. They spin them up on a rig, lap them in and then juggle with the mesh for silence and give a setting for the builder to work to. This gives a silent axles. Otherwise they're noisy as mine is.
ive had my Bentley done this way and it's silent, I had my Bristol dome too and they howl. Mine was silent, but persuading my chum to do another.
I mention all this on the assumption that a similar service must be available over there.