Which parts of a credit card can I obfuscate and still have it be valid

If you deface a credit card, you are likely to find it will be rejected for all transactions. The merchant really needs all the info on the card to be valid - it's part of how they protect themselves from fraud.

So my answer would be: none!

Instead of worrying about that, concern yourself more with how the merchants handle your card. In the UK, for example, a customer never needs to let go of the card in most stores now, as contactless is almost ubiquitous. But if you have to hand over your card, watch it like a hawk. Handheld terminals brought to you are safer than letting someone take your card away.

And remember, if the merchant commits fraud, your bank will reimburse you so it's not the end of the world.

I put a small sticker over the CVV to avoid it being casually seen. The CVV is the three-digit code on the back of the card beside the signature, needed when you buy things on the Internet but not otherwise. A merchant who takes your card and checks the signature can easily remember the three digits, and I think this was what happened the one time my credit card details were abused.

Not having the CVV visible doesn't prevent normal operations, and I can remember it easily. I chose not to scratch it out since it is actually etched into the card, and scratching it to the point of illegibility would have noticeably damaged the card. The sticker would be easy to scratch off if I ever needed to. Nobody has ever asked me to... but I have a European card, I use chip-n-pin, I cannot remember anyone ever looking at the back of the card since the incident where I think my details were stolen.

The better answer is to use Virtual Credit Cards. You can create a throwaway number with a transaction limit and time limit for dealing with each unknown vendor/merchant, you can also set an amount limit on recurring billings. Check what virtual CC setup either your CC vendor (Mastercard, Visa) or issuing bank (BoA, Citi) has. Check the amount of time to setup each VCC number, the ease of setup (website/app?), the minimum time it can be alive for.

Don't expose your real CC number and CSC to untrusted people/merchants. That's a more scalable solution than exposing it. Even now that the US is belatedly moving to smartchip technology, decades behind other countries.