The recipe calls for either, and I have both.
What difference am I likely to see in the final product using one or the other?
Would there be an advantage to going halvsies?
What you really need is "in the middle". Shortbread cookies are not very different from shortbread pie crust, they use the same principle, but a different shape. So the optimal flour would be pastry flour, at 6-7% gluten, which is between cake flour (4-5%) and AP (7-8%).
If you are not aiming for five-star-perfection according to century-old traditional tastes, either of your flours will do well. AP is probably a bit safer, in the sense that the cookies won't fall apart too easily.
shosrtbread cookies are meant to be somewhat crumbly because they have a high butter (fat) content. If you use AP flour they will be a little more sturdy than the ones made with cake flour. Cake flour is meant for cakes and other breads that have to be light and not chewy. That is the basic difference. However, I have made shortbread cookies with totally non gluten flours. I think the suggestion to made test recipes with both flours is a good one.