If you think about it, the dome and the card are not seeing the same light, because the card is flat, and the dome is curved to read the light as it falls on a three dimensional object with a highlight and shadow side. If your meter has the option of a flat diffuser, it should match the gray card, and the flat diffuser is what you use, say, in copy work to make sure a flat object is evenly illuminated.
In a typical portrait or still life setting, where you can control the lighting ratio, I point the incident dome toward the camera.
Now the question is--how do you get the same reading with a gray card? You might presume the incident reading is correct and use it to fine tune your gray card technique. To photograph a three-dimensional object, you often have to angle the card toward the main light, maybe 30 degrees from the lens axis, to get a reliable average of the main and the fill. Experiment with the angle of the card to match it to the incident reading.
Take a look at the diagrams here for Kodak's instructions for gray card use in cinematography--