The 28 Nosler and the RUM cases share a large diameter body. Some actions like a standard Remington magnum action will not allow the cartridge to pass the feed rails.
If you are going to get a used rifle get one with a RUM Remington action, a push feed model 70 or some other brand that WILL allow the cartridge to fit. Wyatt's has longer magazines for each of the actions I mentioned.
Magazine Boxes, WYATT'S OUTDOOR, INC.
An alternative is to buy a new RUM action through Brownells:
MAGNUM 700 LA 300 ULTRA MAGNUM RECEIVER, SS | Brownells
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I find nothing wrong with rechambering an existing 7mm barrel to the 28 Nosler if it is in good shape. Just realize that a factory barrel has the potential to shoot accurately it will generally foul faster than a custom barrel. I use takeoffs often. Of course I also do my own rebarreling and this saves me a ton of money trying new chamberings. It is true, as stated earlier you may not be able to shoot 180s or heavier bullets. I don't see that as that big a deal. The 168 Bergers with their higher velocity will have a flatter trajectory with similar wind drift only loosing a bit of terminal energy when compared to the 180 gr bullet.
If I were to buy a custom barrel I would get one that would finish at 27" or 28".
I say 27" because Lilja has in stock barrels that finish to that length. Otherwise I'd go 28".
Have a triggerjob done or buy a good aftermarket trigger.
Does your gunsmith have the reamer? If not you might want to consider buying one for your needs. You can alter the distance to the lands and specify neck clearance etc to your advantage.
Finally you are trying to minimize your expenses a Boyd Laminate would be an inexpensive way to get a reliable stock. You could drill it out in a few places to lighten it as well.
When this rifle is completed and you have used it for a while, you can always upgrade the stock and get a new bartlein or similar barrel to complete the rifle making it IMO a custom. At that point there will be little difference between your rifle and a full blown custom as they both should be capable of shooting with the same accuracy, except yours will have cost you far less money.
The acquisition of the parts and seeing it come together should be a fun experience. Do the research and ask questions. Hope it works out well for you.