You cut the spigot on your backplate, while the backplate is mounted on the lathe in its normal operating position to ensure concentricity with your lathe spindle.
You bought a collet chuck which should have all machined surfaces concentric.
There are spigots for a reason. Use them is my advice to accurately locate the parts precisely and concentrically.
Cut on your lathe, there will be no run-out attributable to the backplate. If the er chuck runs out, when fitted snugly/tightly to the spigot, it was a carp product.
Ian P has it listed perfectly, for the machining requirements. Follow that and it should work (provided the collet chuck is up to scratch).