The calculator here:
says I need about four days of thawing, and I only have two. Can I cold water thaw it at the start, and let the rest thaw in the fridge?
It's fine to thaw for a few (or several) hours under cold water and then put it in the refrigerator. Just keep the water cold and refrigerate immediately upon removing the turkey from the water. The USDA and Butterball are being extremely overly conservative, most likely because a lot of people are really bad at following instructions. Cold water good - warm water not good
As a matter of fact, defrosting under slowly running cold water for several hours now, and refrigerating until time to cook is exactly what I would do.
I happen to live in a part of the world where I can be guilt-free about slowly running cold water for hours. There is no such thing as a water shortage here, the melting glaciers are extraordinarily huge. Thawing a turkey that way in some parts of the world (and even here in the US) would be an environmentally unsound way to do it. So in those parts of the world, be sure to buy frozen turkeys several days in advance so there is plenty of time to defrost in the refrigerator.
My gut feel is you could go 1/2 way in water (4 hours) then finish in the fridge.
Or go the whole 8 hours in water and refrigerate.
According to a link in the first comment should cook immediately after water thawing. In that case just reverse and start thawing in the refrigerator.