The "fee" you pay for tax refund is determined and charged by the VAT refund company (Global Blue, Premier...) chosen by the individual stores. The VAT forms have the company name, and you go to their counters at airports to get your refund.

One time my VAT refund was done in-house by the shop using a government VAT form, and the shop deducted a few £ from the refund amount.

It's usually at least 4 percentage points, with a minimum charge or a flat cover charge.

Global Blue has an online VAT refund calculator. For instance €200 purchase in France nets you 25.5 refund. French VAT rate is 19.6%

Therefore, VAT included in €200 purchase: 200 - (200 /1.196) = 32.78, you pay 7.28 to the refund company.

Since I don't see (and never experienced) any difference in refund amounts in Euro by choosing different cards, any difference you collect at the end would be due to currency exchange fees charged by the cards.

Refunds take a few weeks to show up, or never.