Hi,
On the product page, it's the setting of the configuration which is used. The setting in the menu item settings is only for the listing.
So no, you can't overrule the setting including VAT for the product page in the menu item.
Besides, even if you would be able to do that, you would have the same issue in the cart module, the checkout, etc.
The solution is not to configure two menu items with the VAT display being configured differently.
No, the solution is simply to configure the "access level" setting of your tax rules in the menu System>Taxes so that the taxes only display for people in the "public" group (ie. not logged in).