It's completely your choice but IMHO, the 338-06 really shines with the 225gr bullets, offered by just about all makers. The case capacity is just to small to make the 250gr bullets perform properly.
Please don't take this the wrong way. I really like the 338-06 cartridge. I have two rifles at present, chambered for the round. Both are very accurate and do everything required of them, as long as I do my part.
The 10% difference in bullet weight doesn't seem to give the 250 any real advantage. There is of course a bit of increased recoil.
The Hornady 225 SP Interlock has been a mainstay for me since 1980. I've tried Speer Mag Tips, Nosler Partitions and some very expensive Scirrocos. All performed about the same at the velocities the 338-06 can deliver.
I found the expansion on the 225 gr bullets at both short and long range to be perfect every time. I seldom recover a bullet from a moose, even at 300yds.
I can't say the same for the 250 gr offerings. I have recovered bullets, albeit usually in the offside hide, even at close range on deer. Not sure why, the bullets were perfect examples of proper expansion. They killed every bit as well as their lighter counterparts.
I just couldn't seem to get the same accuracy out of the 250 gr bullets as the 225 gr.
I used a powder charge of 57.0 gr of IMR 4831 with the 250 gr bullets. It was the most consistent powder I could find at the time for it.
H4831 was good as well. They both required magnum primers to get proper cold weather performance.