It usually more safe to pay through PayPal, because they bring you your money back if you didn't got what you ordered
But I don't know about the privacy
PayPal offers an efficient buyer protection scheme that easily turns into your favour
That said, by card directly, if can be proven the item was not received or it diverging from what was ordered, Visa/Mastercard/Amex offer hidden protection, known as "chargeback". Ask assistance from your bank. The claim will just be longer and less straight forward, but many times just as guaranteed, if rare
Both options are shit in terms of anonymity, unless you create a fake unverified PayPal account, funded by external payment not credit card or bank, but that is a trick which is country specific (regarding lack of verification and transaction limits), you could need a VPN
I emailed RocketExpress to ask about shipping to my country, used the info email address of on their website, the email was kicked back as undeliverable to them. >:/
Sent them an email after I read you. Unlike you, did not get an undeliverable email error back... But also, did not get a reply to my query yet