Respectfully, I've been there, David.
While you are correct, that information is seldom available at the ATS.
It is possible for a delta to provide the third phase, but with the result of a primary neutral current in the order of the primary phase currents.
The secondary effects, if any, are more subtle.
The third phase, supplied by the delta, may be considered as a single phase transformer fed from the open side of an open delta.
The voltage may be lower because of a lower effective impedance of the third phase.
In the event that the primary is fed by a long neutral, there may be phase shifts due to voltage drop on the neutral.
I have installed quite a few three phase ATS's on circuits that shared a wye-delta bank.
With three phase sensing, the systems worked well and did come on when one phase was lost.
When the third phase voltage was supplied by a delta feed-back, when the third phase voltage was high enough for customer use, the ATS would not trip.
When the feed-back voltage was too low, the ATS would trip.
This depended on the pick-up and drop out voltages set on the ATS.
I did have one ATS problem due to a transformer in a three phase bank on the wrong tap.
The voltage on one phase was chronically low.
When the power was lost, the ATS transferred to the generator, however, when power returned, the low voltage on one phase was not high enough to exceed the cut-in point of the ATS.
The set would hang on the generator and not transfer back to the grid.
I could have adjusted the settings on the ATS control board, but I did it right and had the tap set properly on the transformer.
Back to the problem at hand.
There are phase loss relays available starting at around 100$.
I suggest selecting a suitable phase loss relay and wiring the trip circuit in series with the existing single phase sensing circuit.
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Ohm's law
Not just a good idea;
It's the LAW!