What is the most reputable Bitcoin payment processor that has privacy friendly terms and conditions? BitPay and Coinbase are known to have used chain analysis to spy on customers. What are the alternatives for reputable merchants who do not approve of this type of privacy violation applied to their customers?
I'm pretty sure that all crypto payment processors that deal with fiat <-> cryptocurrency exchanges are doing AML/KYC routines including chain analysis. That's not only true for BitPay and Coinbase but Circle et al. too. As soon as you're trying to move value between crypto and fiat you're going to have to deal with different levels of privacy invasions if you're using a payment processor. The exceptions to those KYC/AML routines is either cash <-> crypto transactions (like localbitcoin, mycelium...) or using a crypto currency that can't be traced by (evil) companies like chainalysis.
tl;dr I don't know any non-privacy invading bitcoin payment processor as soon as fiat gets involved.
ps: However I know that there a various people working on new (open-source) cryptocurrency payment processors. I will try to dig out some additional info on those and update my post accordingly.
edit: interesting update on coinbase: https://news.bitcoin.com/coinbase-helps-fbi-shut-kickasstorrents/
You can just accept the cryptocurrencies directly, there are Bitcoin plugins for many shopping carts. The spying and KYC/AML often comes into it because of the fiat side.
You might try Coin Cafe. They do have KYC, but they are smaller so I doubt they have the resources or motivation to track your purchases unless specifically requested to.