The same thing happens if you pay for anything through PayPal, even though the payment is made to PayPal by a charge to your credit card, if anything goes wrong, the credit card company say that their contract is with Pay Pal, not the company you bought from, so your argument is with Pay Pal.
I am not sure how sure in law the agent's disclaimer is. I think the clause about all complaints should go to the owner, may well be questionable and if you queried it and threatened to take them to court, they might offer you more.
Having said that when a retailer, whom I paid direct with my card, failed to supply the goods, I went to the card company and they reimbursed me.
It is just that they only repay on payments where they have a direct relationship with the company owing the money and, if you can step back from your present problem, you should be able to see that this is not an unreasonable demand. It is up to the purchaser to be make sure when making purchases to check whether they are paying an agent who filters the payment to the service supplier or the service supplier themselves.
I never use PayPal for payments of over £100, for just that reason.