which pin on the card connector (1, 5, or 13) would be the ideal place to connect an internal shield or ground connection of a transformer?
I am going to disagree with the other posters who recommended connecting internal transformer shield connections to the chassis connection.
The purpose of chassis is to keep EMI
out of the inside of the chassis, and by connecting an
internal component shield to the chassis, you have now made a connection between the outside and inside of the chassis, which bypasses the point of having a shielding chassis to begin with.
To determine where the best place to connect an internal shield would be, you need to determine where the circulating currents would be sourced, and make the connection which returns those currents most directly to their source.
That brings up the exception to the rule I mentioned above, which is EMI filtering components. Those might technically be inside the chassis, but the source of the current is coming in on the external wires, so the filtering components should be as close to the input connectors as possible (ideal would be
on the input connectors) and return to the chassis connection.
So the first question to ask is what is the shield shielding (i.e. what is the sensitive node which is being protected), and what is the shield shielding the sensitive side from (i.e. where is the source of the interference)?
In the case of an input transformer, or possibly an output transformer, you may decide that the shield is actually part of the EMI protection, in which case it may be appropriate to connect to the chassis connection. The best solution in those cases is dual shield windings, one around primary and one around secondary. You see that in some Jensen input transformers (note also that the Jensen diagrams indicate whether the shield winding is closer to the secondary winding or closer to the primary winding by the way the symbol is drawn).
In other cases you are trying to prevent e.g. power supply noise from getting across from a higher voltage winding to a lower voltage secondary winding, so the primary winding is the driven side, and the transformer shield winding should be connected back to the ground for the primary side to return parasitic currents back to the source.
In the example screenshot you provided, it appears to be a signal transformer with 1:1:1:1 windings, so it is not clear how the transformer is used, what is driving and what is receiving. If the two sides are truly galvanically isolated it can be better to tie the shield to the secondary side ground reference, because if you tie it to the primary side then any common mode noise (i.e. primary side ground reference moving in relation to the secondary side ground) will be imposed on that shield winding, and couple through parasitic capacitance into the secondary side windings.
Sorry the question is not quite as straight forward as you were hoping, but for best performance the details matter.