You are mixing some things here.
This would be a separately derived system and needs its own ground rod(s). Tie into existing ground only to meet common service requirements. The key word here is separate. Treat it like the primary side is a “utility”. As such you fuse or breaker both poles on the secondary. There are engineering arguments about common ground grids but NEC stresses separate ones. One of the issues is you are creating another path to ground through the new transformer rather than back through the system bonding jumper which can only exist in one place at 120 V. Another is since they are not isolated surges on the higher voltage side affect the lower voltage side. I’m guessing though you are going from say 480 or 240 to 130, not say 4169 to 120. The easy (and cheap) way to do this is an auto transformer where by nature your neutrals are automatically “jumpered” since it is the sane terminal.
Jumper from the primary side neutral to the secondary neutral. Keep grounds separate, IF you have access to a neutral. Otherwise see above. This just makes it an extension of the higher voltage system. Think of how an AC “wall wart” works...the load is just a load even though it’s behind a transformer. And we carry the bonding through as a continuous isolated system.