Lest anyone interpret my opinion about "loose legs" as a causative
element of piano shimmy, let me hasten to say it is neither definitive
nor absolute, as are many aspects of our always-interesting avocation.  
It is, however a pretty reliable indicator, and one that can be fairly
rapidly analyzed and corrected or eliminated in the search for a cure.

The root cause is likely to be an out-of-balance condition in the
kinetics of pump operation, as others have observed.  It is likely that
the designers, installers and early maintainers of these systems gave
little, if any thought about "operating balance", although it can be
argued that they should have.  The malady seems to arise infrequently,
which indicates to me that the resultant lateral forces on the piano
case are relatively benign.  But a more pronounced imbalance, coupled
with compromised glue joints in the legs can certainly lead to the
condition.

Personally, in my experience, I have never observed a pump to which any
balance-corrective measures have been applied, either by the addition
of strategically placed weights, or the removal of mass from moving
components by drilling or grinding as is typical in many mechanical
balancing techniques.

As to the source or root cause, since the (typically) four exhauster
bellows have mass and moment arms (probably)within a few percentage
points of each other, and the "spider" has even less mass/moment,
I can see only two other sources of imbalance force(s).

The first would be inconsistency in the tightness of the flap valves,
whether internal or external.  This condition is not easily diagnosed,
and one where it helps to be, or have access to, a pump "whisperer".
In the event that similar inconsistencies occur in two bellows at right
angles to each other, the effect may be additive.  If they are across
from each other, the forces should tend to damp each other out, but
there is no guarantee.  The other source would be frictional
differences between the bearings of the connecting rods where they
attach to the bellows and spider posts.

I'm going to go out on a limb and say that the pumps of shimmying
pianos are more likely to have metal connecting rods with ball bearings
and "cone" posts and nuts, and that improper or inconsistent adjustment
of these connectors can lead to widely differing frictional differences
such that the pump _wants_ to "lope" like an air motor with a "set",
but since it is forced to turn at a constant speed (rpm) by the
synchronous motor (unless the belt is slipping), _then_ the frictional
imbalance induces the lateral, pulsating force(s).

It would be interesting to confirm which type of connecting rods are in
the pump of original inquiry, and whether there are any known instances
of case shimmy in Welte Licensee (or similar) pumps with flexing fabric
connecting "rods".

As far as trying to "balance" a given pump without giving attention to
either the flap valves or bearing friction sources postulated, I would
suggest taking an approach similar to that used to balance the blades
of a ceiling fan.  After making sure the legs and leg joints are as
tight as possible, obtain a suitable weight, perhaps 8 to 10 ounces
to start, and affix it in the center of one of the flywheel spokes,
perhaps with a worm-drive hose clamp, and observe whether the shimmy
increases or lessens in amplitude.

When you have found the spoke position that makes the best improvement,
then move the weight closer to the hub or further away.  If the best
improvement is at the hub (but still noticeable) start over with a
lesser weight on that same spoke.  If the best improvement is at the
rim (but still noticeable), start over again with a greater weight on
that same spoke.  Hopefully you will find a weight/position combination
which will minimize the shimmy and not be too unsightly or rub on the
belly cloth, in which case, you are on your own!

John Grant