well to start with for a printer that size - you need at least 10mm smooth and threaded rod, for a cartesian I'd go for 12mm.

Those are long rods carrying a fair bit of weight.

A skr 1.4 turbo board with 4.5 inch screen and tmc 2200 stepper drivers will do everything else for not a lot of money.

But do not skimp on the frame and mechanical side of things.
Your carriage support rods need to be completely rigid and on a 500mm print volume, that's - at least - a 600mm long rod. So 12mm would be the minimum I'd consider.

Price is not an issue: obviously, the lower the better.
Well, price is an issue and the lower the better will not make the best printer.
There is usually a compromise area between teh most expensive and the best value.
Either way you almost never end up using the lowest price parts :-)