What are the chances that the Razer core would work with the Gigabyte P34W v5
From 0 to 100%. You can check with Razer support.
As of March:
"n terms of compatibility with non-Razer laptops, Sather said the Core is designed to be open to any laptop that can support it. That means that a laptop needs to fully support the external graphics standard for Intel's Thunderbolt connection.
"It's on the bleeding edge of these standards," Sather said. "Anyone who adheres to it should have a laptop that supports it." The only system Sather knew of that does support the standard is a recently announced Intel NUC mini-computer."