Lets price this out to see what it will take.
Surgeon single shot action $1350 (Stiller TAC 338 or Phoenix $1000)
Manners stock-$475-675 depending on model (Russo A5L $300+ unfinished and inletted, add $300 for final finish and clear coat)
30-32" Barrel (requires 1.350 shank for surgeon and others) extra length & oversize diameter blank will be $375-400 minimum depending on mftr
I have a 1.350" 338 barrel (fluted) on order from bartlein for a 338/408 and it is $565
Chambering $175-200, can go as high as $400 depending on smith
Brake (new Holland Radial $105 is absolutely awesome on my 338 AI) another $100 installation
Pillar and final Bedding $200
Trigger w/safety 1.5 lb range (Jewell $225 Rfle Basix- $125)
flush cups, sling swivels and installation $100
Misc gunsmith (trust me it will be there) $100-200
Chamber reamer and go/no go gauges ($200) if you want to keep them and rebarrel in the future with the same chamber dimensions and dies.
Dies (338 AI only choice is custom from the smith from his reamer set (chamber and resize reamers required) or Neil Jones set with 3 shoulder bump bushings and micro seater, got my Neil Jones 338 AI dies yesterday for $366 and worth every penny.
Bottom line is that if building a new gun from scratch, $3000 will not make it normally much less leave anything left over IMO unless you sell your gun and apply that with the $3k.
You really need to decide first your gun or new gun and then start pricing components out and seeing if they fit in the budget. I am not sure if the MK5 can/should be converted to the 338, particularily the AI version. I am not saying it cannot, I just do not know. It might be able to, I would do some checking first and be totally comfortable with the answers. For example do you want a magazine gun or single shot? Can the MK5 mag fit the 338 AI? If you want a magazine gun, look at the 338 Norma Mag (new Norma case that is basically shortened Lapua that fits standard long actions, but gives Lapua ballistics). Brass and dies are available. Long thread on Snipers Hide about the 338 Norma.
Also do not forget the 338 Edge which could be done for about half your $3k budget with a barrel, brake, trigger and stock. 300 gr SMK at 2800 fps is nothing to sneeze at. Shawn has the reamer and on shelf dies, so rebarreling later is super easy and will match the old one.
The 338 Norma could probably be done to the Mk5 for well under the $3k. Plus the 30/338 Norma with 240 SMKs is a strong LR contender.
Look at the Snake 2, that I just built. Snake 2- 338 Lapua Ack Imp
After completing fireforming the brass it is going to be disassembled and the bolt and barrel are going to be melonite coated (super hard industrial coating to hopefully significantly decrease throat erosion) $200. Only way I got it together at that price was I got a deal on the action, stock and barrel from guys needing to sell them and saved a lot of money.