If only the reverse lights are an issue and your reverse lamps are only
for reverse, I'd attempt using a simple relay off the reverse signal.

No it does not.  Did the scantool also at the same time show the switch
input from the gear selector switch to be working?

The explanation your being given makes no sense given how computers
are designed.  Your computer uses a power circuit that drives the
reverse lights and carries a lot of power.  If there is going to be a
problem in the computer it is most likely to be here, where the
higher power components are most likely to fail.  In that case nothing
they did with the scantool would make the rear lights come on.

By contrast the gear switch input to the computer is either open or
closed.  If the sensor wire were to be crushed and shorted out, it would
be the same thing as closing the switch contacts - it would not harm
the computer.  If the sensor wire was to be crossed to a hot lead in
the car it would be the same thing as a logic high - electrically the
same thing as an open contact to the computer.

I thought in any case with that vintage that the transmission computer
used a fully electronic gear selector.  If the gear selector switch
was broken then how would the vehicle even go into reverse at
all?

You need to bring it to a shop that specializes in automotive
electronics first.  I am not sure how the trans computer communicates
with the body computer I had thought it was by the shared bus.
The diagnostician needs to jack into that bus and see if the
trans computer is showing a reverse indication AND more
importantly, that for that year that the body computer is even
using that indication, and does not have a separate input to
a reverse switch somewhere, such as inside the transmission.

If you find one in a T&C with exactly the same options and programming
as yours, there should be no reason you can't use it.  That includes exactly
the same anti-theft programming of course.  But, if it was my vehicle and
it really was the computer I would not touch the computer.  How often do
you use reverse?  Very seldom.  How often are reverse lights even
seen?  Far less seldom than that.  If it were me I would go buy one of those
truck backup alarms, and wire it and the reverse lights to a toggle switch
near the steering wheel.  In the few situations where your backing into
traffic
where the traffic can actually see your reverse lights, then use the toggle
switch.
You can drive, can you not?  If so, then remembering to flip a toggle during
a backup manuever should be well within your abilities as a driver.

Ted