6.5mm would be suffcient and you could use 7007 bearings, but not 7008 for the reasons given earlier. There's no problem in terms of strength from using the smaller bearings, since compared to the bearings you'd use if spinning the screw the ratings are still far greater. Don't even think about deep groove bearings. They're not designed for axial loads. When you put an axial load on one, the balls are forced to ride up on the side of the ring, which isn't ground to such a good finish, so they wear out very very quickly.

How much have you found that spindle mount for? I could probably get a block of aluminium and bore it for you for less money...

Have you thought about how to make the shaft? I'm guessing your lathe is big enough, but bear in mind it needs machining very accurately for the bearings and nut with good concentricity. It also needs thread-cutting for a locknut.