I just went through this myself and asked all these same questions a little over a year ago. Here's what I learned and what I ended up doing:
Zermatt Arms/Bighorn Origin Action, Grayboe stock, Trigger Tech Trigger, Proof Barrels, Hawkins Precision BDL bottom metal, Leupold VX-5: 8.5 lbs total
1. Budget and Timeline: Your budget is excellent and very realistic. You should easily be able to stay at or just under $4k. My project basically took an entire year because of how long I had to wait for the action (several months) and other components (the trigger was easy but I had to wait some time for the stock and bottom metal too).
2. I worked with a Gunsmith for all the reasons stated above, but primarily because of my own personal lack of knowledge and experience in building a traditional custom backcountry hunting rifle while trying to keep weight down. Now that I know what I know, I might do exactly what
@brcfo_outdoors suggests but I found having a trusted smith there to ask questions of and get advice from incredibly valuable. I wanted to source all of my own components because I love learning about everything but I'm also particular. As I did research and had questions (particularly on action choice), my gunsmith was always there with excellent perspective. I worked with Chris Benfield at Benfield Precision in Idaho. He only takes on a few projects each year so that he can focus on your build. I sent all my components to him as I purchased them but he was more than happy to do all that work too if you don't want to.
3. Action choice: This was the hardest part for me. Because I wanted a dedicated hunting rig, I eliminated all the very expensive actions and I didn't feel titanium was worth the cost to save the weight. If I had unlimited funds or didn't care about cost, I might have considered it. I knew I wanted the option of changing barrels and having the flexibility to explore other cartridges down the road. I didn't want to have my gunsmith do too much custom work or milling on the stock so I looked at actions that were cloned or at least based on the Remington 700. For around $1,000 I had two excellent options that I narrowed my choices down to: the Zermatt/Bighorn Origin Action and the Kelbly Atlas. There are differences between the two for feeding and ejection but both are excellent options. I chose the Origin in a long action because I liked the controlled round feed, manual ejection, and because I knew I would want to swap barrels between .280 AI and .300 Win mag. There are great options for prefits for me to play with in the future should I want any other long action caliber to
@brcfo_outdoors points above. Getting yourself a barrel vise and good torque wrench is easy and Zermatt will send you the action wrench. The Kelbly is also a great choice for switching barrels and you can find prefits for them too (although I think Kelbly is no longer making them for their actions in house). The difference there is that I would have had to buy an entirely separate bolt to go between standard and magnum and Zermatt has a clever way of easily swapping the bolt head out (it's too-less and only takes a minute). Far less expensive than an entire bolt, but not so much that I could easily dismiss the Kelbly. Ultimately for me it was a combination of cost (Atlas is a $200 more plus the added cost of buying a second bolt), barrel choices, and my preference on feeding/ejection mechanisms. Weight differences between the too were negligible.
4. Stock: I spent some time on this one too because I wanted a traditional sporter stock profile and I like the dry and textured feel of composite/kevlar fiber stock over the slippery carbon fiber stocks. I had to sacrifice some weight because of this but I am very happy with the Grayboe Outlander and their quality is incredible. They also have a great website and ordering system with all the detailed inletting options ready for you to choose from so your stock arrives truly custom fit and ready to drop your barreled action into.
5. Barrels: You can't really go wrong here as long as you are selecting a quality barrel maker. You'll have to decide if you want carbon fiber or stainless (you can have it cerekoted to any color you like). This is another area where your gunsmith will be invaluable and likely has personal preferences.
If you decide to not go with a gunsmith, another option would be purchasing a barreled action. Altus, Patriot Valley, Southern Precision, and Northland all sell these. Then you can just source your trigger, stock or chassis, and bottom metal to build it yourself. A gunsmith is still good to work with here just to make sure everything is timed and your trigger pull weight is where you want it assuming you don't have a gauge of your own (they're not expensive though). At the very least, you might find peace of mind by paying a good smith to look over your work. If you're going to get into hand loading seriously and you probably should if you're set on that caliber, having a gunsmith chamber your barrel to your action is still a better option. They will have the proper reamers to ensure you have the space you need to experiment with the Cartridge Over All Length (COAL).
If you want to shave a few pesos off the project and don't care about getting your accuracy under 1/2" MOA and choosing every single component, there are some excellent semi-custom builds out there with complete rifles for sale. I like what the boys over at Red Hawk Rifles (also a good resource for buying components and actions) are doing here:
https://redhawkrifles.com/proof-research-5/ Your issue will be finding a complete rifle in that caliber - it's probably not popular enough to see in some of the more affordable options like this Red Hawk Proof rifle build. You can go to Gunwerks or AllTerra and get one but they are expensive and I think you would enjoy building your own more. You'll have more control over everything and you'll stay within your budget too.
It may not be custom, but if your budget shrinks, it's worth looking at Bergara Premier, Christensen Arms, and some of the other production options out there. A good gunsmith can also work wonders with a Tikka T3 donor rifle. You could source a nice used one for like $650 and have your gunsmith customize everything else for you. Even with chambering a match grade barrel and upgrading trigger springs, the bolt stop, bottom metal, etc. you're probably clocking in around $2,000 before adding the optic in.
Good luck! You'll have fun no matter what but if you want it for the 2023 fall hunting season, you better decide now - even then timing could be tight unless you find someone with the action you want in stock.
