Is it possible to allow the user to use Budget and pay by Bank (CC/DC) Cards both?

No, it is not possible to allow the user to use Budget for some and pay the rest with bank card at the same time for the same order. Budgets are not a payment method.  They are a spending limit that applies to all available payment methods. The payment method determines how the user will pay for the order.  The budget determines how much the user is able to spend within a given period of time.
It is possible to allow the user to checkout without using budget through the site group permission “Can Checkout without Selecting a Budget” however the user will still require to select the Payment method at the Checkout Page. To allow the user to pay partial amount using Budget and pay rest using bank card is not possible.