I was trying to find if NASA did any zero-G mass measurements prior to 1985 and came across this:
Dr. Thornton returned to active duty with the United States Air Force
and was then assigned to the USAF Aerospace Medical Division, Brooks
Air Force Base, San Antonio, where he completed the Primary Flight
Surgeon’s training in 1964. It was during his two-year tour of duty
there that he became involved in space medicine research and
subsequently applied and was selected for astronaut training. Dr.
Thornton developed and designed the first mass measuring devices for
space, which remain in use today.
At first I couldn't find any description of Dr. Thornton's measuring device, but Heinlein's "elastic inertiometer" is a fairly obvious, though still elegant, solution to the problem that surely a scientist like Dr. Thornton with a BS in physics would have come up with on his own.
Coincidentally, the SLAMMD (Space Linear Acceleration Mass Measurement Device) I mentioned in my comment is used primarily to measure the body mass of astronauts to study how zero-G affects astronauts on the ISS, and this is also what Dr. Thornton specialized in as an astronaut.
And I eventually found this patent application for a "NONGRAVIMETRIC MASS DETERMINATION SYSTEM". Its description:
A non-gravimetric mass measurement system having a support structure
for holding a mass to be "weighed" attached to an oscillating spring
assembly. A device for indicating precisely when the oscillating mass
crosses point of zero displacement produces a signal which is sent to
a counter for determinating the time period. A substantially
frictionless air bearing serves to restrain the motion of the
oscillating mass to a single axis under zero gravity conditions in
space while serving to support the weight on earth.
This patent was issued Jan. 19, 1971. It was filed May 20, 1968, so quite a bit before the "Cat Who Walks Through Walls" was published.