A problem on the secondary side will never cause the GFI on the primary to trip. There is a problem somewhere in the GFI circuit on the primary side. There may be another plug that is on that circuit without an in-use cover or a nick in the wire somewhere.
This is a good reason to have a dedicated circuit for the low voltage lighting. You don't really "need" it for 650W of load but if you had it, a problem like this would be less likely to happen and far easier to troubleshoot.