Which Earth should a chassis be connected to: supply, or local?
IMO, both your cases are incorrect.

the connection between a provider distribution grid as the supplier and a house (home/building) as the consumer typically looks like that (remember, at least one full, i.e. L+N, breaker should be installed after the feeder enters into the house and before the in-house wiring/distribution):


by this, earthing the in-house apparatus should be like that (remember, you need to use a multi-pole [L+N] breaker before each critical apparatus, e.g. xformer or welder, instead of a simple one-pole [L] breaker if you cannon ensure the direct wiring, i.e. L to L, n to N, not L to N, will take place all the cases and times):


and last, if you decide to place an insulating xformer (by some your reasons which are not clear from the question) that could be if the distance between the feeder input and the place is too long to the PE wire to act as a protection correct, its case (if any) should be earthed (grounded) locally as well as the new PE wire going to the apparatus in the new insulated domain (remember, the L+N breaker mean also rules here):



Therefore, my advice for you is the next:

provide you apparatus with an electric plug wiring out L, N, and PE (connected to the case, if any and conductive) circuits;

provide you xformer with an electric socket compatible with the plug on the apparatus, with its PE connected to the locally grounded circuit---at the secondary side (winding), and with an electric plug or terminals wiring L and N, but not PE, compatible with the socket or wires on the in-house power wiring---at the primary side (winding).


Good luck.
P.S. Above is related to TN-C based distribution grids typical for most Europe, imo.