Let's get back to the OP. I think if you're going to have a custom build - have a true custom build. Half the fun in doing a custom build is the research for parts, and the expectation of how sweet the end product will be. You state that you are left handed, then buy a left handed bolt action. Kelbly was mentioned, I have one, and can verify that they are smooth and strong.
If weight is an issue, why go with a #5 bbl? Especially if it's 26" long. If you've never carried a rifle with a 26" bbl, there 's a big conveience difference between that and a 22" bbl.
I have two suggestions for you. First, make a list of what you want for an end product. A left hand bolt in some form of 7mm Sherman (if you are dead set on that cal, altho the 338 sounds like a good idea to me), and its use, are the only specifics you need to list. After that, list a weight range for the rifle, an energy/velocity/BC range, approx cost, and anything else that is important to you. Then, instead of going to a forum that gets bogged down in petty BS, start calling acton manufacturrers and see what they have to offer, in both products and opinions. Most of these people will build a complete rifle for you and it will be a much better product than rebuilding an over-the-counter rifle.
The second option is to take your list and do the leg work yourself. Let Google be your friend. Just Google " bolt action receiver manufacturers" and be overwhelmed. lol This is a better option than you may think, because it will expose you to options that you never knew existed. It will also give you more options to give to your gunsmith.