They’re not refusing anyone where I live in New Orleans based on residency or citizenship. Whether they’ll supply you with documentation that acceptable or not, I have no idea.

Your best choices I believe are (a) to try to reschedule the vaccine earlier in the EU, or (b) delay your trip until fully vaccinated in the EU, or (c) move your trip up so it falls between doses 1 and 2 in the EU.

Property tax receipt/deed plus utility bill both in your name(s) should be fine.

At this point, in my area of Virginia that has 90%+ uptake of the vaccine, if you have a pulse, you can walk in and get a shot.

I imagine it's pretty much the same everywhere in the US.

Typically when you move your vaccination status moves with you. E.g. a kid moving from the US to EU will have their vaccination schedule copied in the system of the destination country so the health authorities can ensure vaccines are current. I suspect it is exactly the same in the US. FYI - getting the 2nd dose in the US is easy - getting the 2nd dose in the EU if it is not yet your turn - after receiving the first dose in the US is difficult - as most systems are not set up for exceptions and/or have limited supplies of pfizer/moderna.

FYI - Flying transatlantic is the safest it will ever be. Flights/airports are near empty and all passengers are tested (sometimes twice) - with likely many already vaccinated.

If the vaccinating site has staff that thinks you're under 18 years of age, you may want ID that shows that you're at least 18 years old or older. Even for the Pfizer vaccine which is approved for those 16 years and older.

In California the only requirement is that you roll up your sleeve.

They'll ask for ID and an insurance card, but they'll also have signs up that say it's not required, and if you have a non-US ID, that's ok, too. It's so they can keep a record of your vaccination and bill your insurance or the government as appropriate. Mostly the goal is to just vaccinate everybody. In addition to my CDC card, Walgreens has an online record in my account of the vaccine so I can display it on my phone at any time.

I realize the CDC is an American organization, but they say the Pfizer second dose is valid up to 60 days from the first. Depending on the timing of your stay, perhaps you can get the second dose on your return?

Florida requires residency documentation. As far as I can tell, most days you cannot count on just walking in (even if you qualify); I asked out of curiosity at my local Publix bring firmly told "Appointments only!" I asked the person shooting me up at Walgreens: Some days we have extra, some days we don't. Haven't made the call on today specifically.

Having a requirement for "appointments only" isn't the same thing as being required to show local residency documentation. I have no doubt that Publix stores selling groceries to people who fly into SRQ and TPA have been providing vaccinations to people with booked appointments even when the people flying in have no local residency documentation.

0000 for the SSN field is what has been done for people who don't have a SSN when making the vaccination appointment at Publix. [Not all US citizens have a SSN.]

While Florida tries to claim that vaccinations are open for people who are residents of Florida and non-locals who are providing goods or services in Florida, in practical terms vaccinations are being provided at Publix sites and elsewhere in Florida also to foreign tourists.

[Receipts from foreign tourists in Florida get classified as US export receipts, and export receipts come from the sale of good and services.]

Florida does not require Florida residency documentation to be eligible for vaccination in Florida. Florida says the following are eligible for vaccinations in Florida: people who are residents of Florida; and people -- regardless of residency -- who are providing goods or services in Florida.

There’s a wide variation in what various states say their rules are, and what happens when you actually show up onsite. If you get denied in FL, just drive or fly to Mississippi or Louisiana and try again. If you’re willing to travel a bit- you can get the shot.

Or just walk or drive around to a different vaccination site in Florida. Or even just come back to the same place later the same day.

Florida has confirmed that as of 29 April "all state-run and federally supported vaccination sites" no longer require any proof of residency in order to get the vaccine.

As to the implications asked by the OP, I suspect keeping the two separate paper proofs of each dose of the vaccine will get you through 95% of situations where proof of vaccination will help you out. Keep in mind, there is no agreed "standard" of what constitutes proof of the vaccine in Europe (as of today). When you get back to your country of residence in Europe, see if they will update your health records with the dose given in the US based on the CDC card. If not, ask them what proof they would like. The CDC has indicated that states are to be keeping digital records that can be requested as more "official" proof. I cannot imagine any competent health authority in Europe (or your GP) will not accept one or the other of these in order to update your local health records.

At the end of the day, I suspect NOWHERE in FL (and likely many other states in the US) is going to absolutely require proof of vaccination for anything. In Europe, it's likely to make life easier throughout the summer, but in most cases there will be some alternative to proof of vaccination that will suffice until you get it sorted out. While the situation described here is "unusual" I doubt it is going to be "extremely rare" and authorities on both sides of the pond will find some way to address such instances.