Would I have a problem with extra thin necks from necking a .375 neck to .458? I would much rather neck down brass and outside neck turn for uniformity, but I have not found RUM basic brass yet. Not unless anybody knows who makes the RUM basic stuff...

Yes, I do know that the RUM design is slightly inferior due to the rebated rim, but the case head is fatter than the actual .404. Thoughts?

Gieter Horneber will make basic for you. If you buy enough (I think it was 3k, but it might have been 500) he'll even refrain from rebating the rim.
The original Lott was made from 375 H&H brass, so it can be done. But if it were me I'd use the Jeffery. The extra 25 fps or whatever is simply a non-reality outside of the paper world. Jeffery brass isn't all that expensive when you consider 100 may last you and your son. Or if not, you know the brass will still be there when he's 45 and looking to buy more.
Vincent got his 458-404 over 500/2500 fps IIRC, so it will do what you want it to.
Also, keep in mind, bigger casehead = more boltthrust, which if you're talking Mauser/Winchester/M1999/Ruger might mean hanging out around the topside of max. If you're looking at the MR PH or CZ Mag then shame on you for not using the Rigby case (and a .475" bore!!)

Well, bolt thrust really would not be an issue. I have been entertaining the idea of getting a Ruger #1 and having it rebarreled. I think that might be the best bang for the buck as compared to spending 3 to 5g on a super nice bolt gun with a Granite Mountain receiver. As far as the RUM brass goes, Ruger does not offer a #1 with a RUM chambering so the cost savings on brass would be offset by the gunsmithing on the receiver besides the rebarrel job.

Hmm. For the #1, you don't need it chambered in the case, you need it chambered in the caliber. So a 458 Winchester would do just fine.
Now, you kinda lost me about the price of brass; not sure which side you were arguing for, but what I was trying to get across was that when you look at how much this whole thing will cost you, buying 100pc 404J brass will add about $90 to the project over using RUM brass, and you'll have the security of knowing the brass will always be there.
And as for bang:buck, you can get a M70 in 300 RUM for not much more than the price of a Ruger #1, and add a barrel for another $175 or so, or just get a CZ in 458 and have the feed work done, and still be in the same ballpark as a custom-barreled #1. So there's a lot of room between a #1 and a Granite Arms full-custom.