For the scope I'd go with a Nightforce 5.5-22x56 (or 50 ) with the NP-R1 reticle. Why? Top quality, rugged, repeatable clicks, super glass and the R1 reticle is great for either dialing and shooting, spotting your own shots and knowing the error, if any, and ranging if needed. Use the best rings and base you can find, with the base probably being 20moa or more. With the 5.5-22 NF you can get a loooong ways using the 20moa and the generous internal clicks.
The 338 AM will need a custom action, the Edge can be built on a Rem 700 or similar. Big difference in price there. Also if using the Rem you can count on having it accurized etc. Hang the best trigger you can afford on it.
Shoot the best bullets the gun will handle accurately out of the best brass you can get your hands on.
If it's a dedicated long range rig I'd start with at least a 30" barrel but with the AM I'd go longer. Brand is up to you. There are many good ones out there. Consult with your smith and see what he has had the best results with.
Stock choice is up to you and your preferences. I like the A5 design although the wait is a pain in the butt. Manners, Joel and others can get you similar but much faster.
Add in a scope level of your choice and probably a ACI along with a Kestrel and Pocket PC or Palm running Exbal or similar and you can start having fun.
All of this is just personal preference and yours, and others may and probably will vary. In the end you just have to make you happy.;)
Get everybody's suggestions, weigh them, decide what you personally want and then contact your smith. You may find that in talking with him some choices may change. The two calibers you mention are usually connected with two of the best smiths out there. They know their business and will help you with every step of the build.
Good luck and have fun.