Most digital printers use ink that is effectively transparent. You can see the background. If you are printing on blue stock, it will not show up very clearly, if at all. If you want it to show up, you need to use foil.
If you need to keep costs down I would use white card stock and print the blue background, with whatever else you need knocking out the background. You can print the background blue on the reverse to suggest blue stock, but the edges of the card would be white.
350gsm is standard for business cards. That might be too much of a step down from 540gsm previously used. Digital is quite capable of printing on 400gsm stock, you might even be able to find a machine to print on 540gsm stock.
This is true but not many digital printers will do this. Also, it is only feasible (as you alluded to) if you want white printing - using it as a map for the background to then overprint something is not recommended.
Digitally printing on 350gsm “color plan imperial blue” stock with CMYK gray can produce clear results, but it may not have the same metallic shine as foil. If you are budget-conscious, consider printing digitally on white stock with a blue background, as this can be more economical. 350gsm is thick enough for a premium feel, but you can also explore 400gsm navy stock for a slightly thicker option. Your previous 540gsm with frost foil had a luxurious appearance, but it comes at a higher cost. Your choice depends on your budget and the level of luxury you wish to convey.