the problem with this method of purchasing bitcoin is that the sellers have to worry about fraud. the stolen cards that can mean the money being charged back (reversed) and since bitcoin has no charge back they have to take the precautions and ask for verification.
the ATM suggestion seems the only left method but it won't be "cheap"
This is why 3D Secure has been created. So Every merchant can set up this for purchases using a card. If the card is stolen, you won't be able to bypass the 3D secure process.
The problem, as the OP wants to buy "cheap" Bitcoin, is that merchants are charged a fee for using the Visa/Mastercard network and so they forward those fees on the price/you.
The OP can use Paymium, they don't ask a verification ID up to 250$ bought with a card