Please don't make customers not having Bitcoins also your problem, chances are if they don't have Bitcoin in the first place they will just cancel their order and won't go with the extra effort/hassle of buying BTC and paying it off with their credit cards just to continue with their order. As for the Bitcoin only payments to your website the easiest way to integrate Bitcoin payments is to a payment processor like Coinbase or Bitpay as they will provide you step by step on how you will integrate BTC payments for your site. Just promote your website as a Bitcoin Only E-commerce site so you don't have the wrong audience shopping in your website.
Agree with promoting it as a Bitcoin e-commerce website. True, you might lose out on fiat customers but it appears you're here and want to focus on Bitcoin users.
In addition to the payment processors above, I would suggest Mistertango as an alternative. If you're willing to get some elbow grease into it BTCPay is an open source solution that I've seen some merchants implement with success.
Benefit of the latter is more control over how you accept payments, plus you no longer have to bend over for the changes Bitpay and the like seem to impose with increasing regularity on both merchants and consumers.
