The only alternative to traditional car rental or the train I can think of is one of the 'free-floating' car rental services that exist in Antwerp, provided they'd let you create an account without a Belgian ID or driver license. They don't explicitly mention it as a requirement, but you'd need to check.

GreenMobility would probably be your best bet as they explicitly state that you can use (but not return!) the cars in the Netherlands as well and they offer some packages that might fit your needs. E.g. 1 day at 75 EUR or 2 days for 125 EUR.

Mind you, the cars are all pretty small and they're all electric. So while you don't have to return it full, you would probably need to find a way to charge the car (free at many chargers in Belgium, but at a cost to you while in the Netherlands).