I calculate a minimum of 0.226", based on the case dimensions I found in Patagonia Ballistics LoadBase 2.0 Library. The exact measurement would likely be a little greater because many chambers have some additional neck cut into them, above and beyond the minimum SAAMI required in order to accomodate some excess overall case length.
I get a case neck length of 0.306" for the 338 Edge. And a maximum case length of 2.850" for the Edge, and 2.770 for the 338 RUM. The longer 338 Edge case length will get you 2.850-2.770 (0.080") into fresh throat of the 338 RUM chamber. But you'll still need 0.306 - 0.080= 0.226" of additional fresh bore for the remainder of the 338 Edge neck length, which will only be obtained by setting the barrel back that distance. And on top of that, whatever additional neck was reamed into your specific 338 RUM chamber, above and beyond the minimum SAAMI requirement.
I see you've posted in the Gunsmithing section. Just to let you know, I am not a gunsmith. Hopefully one or more of them will sound in. If you're serious about this, I think I'd talk to the smith you're planning to have do this work for you.