Under $4k budget build LA360 build

the machine work will cost ya. as in cylinder boring honing heads redone etc. crank polishing etc. etc. I used to "overhaul" engines foe under a grand No machine work new bearings rods oil pump all gaskets. new cam etc. But that was years ago. maybe 20 years ago I did a 318 like that, what pushed it over $500 was that I bought a new Edelbrock aluminum intake that is now on my 340:) basically you'll do what the budget dictates, and take pistons, if they are used and can be cleaned up, why buy new ones? Its all up to you and your wallet. also an eye for inspecting the parts. I will admit I did put new pistons in my one engine because I has TRW domes in it and wanted to lower compression. But not necessarily because the pistons were bad.
it sounds like you are on your way though. Build your engine and drive the car meanwhile, then swap the finished engine in one day and fire it up. Set the other one in the corner and go over it at a later date. thats how it goes. they say a true hot rodder always has a spare engine sitting around:)