How to handle a paper by a reviewer who wants to be paid?

First of all: Is there a rule that you need to be available as reviewer if you want to publish in that specific journal? People decline reviews for all sorts of reasons (often lack of time, which may or may not be the case), and I never heard that somebody got "punished" for that.

Your reviewer wanted to get paid. It is very unusual to pay reviewers, but on the other hand, it is not per se unreasonable or offensive to ask for compensation for work.

I would see it this way: Either you establish a general rule that authors must be available as reviewers or you accept that some people do not review papers (although they publish).

The editors of a journal should be professional at all times. Like in any part of life, in academia there will also from time to time be annoying individuals. Don't get down to their level.

Your point 5. would be simply acting out of vengeance. Regarding points 3. & 4. – is a fee a regular thing for this journal? Because if not, it would be a vengeful misconduct. If there is a fee, is it usually used to pay the reviewers? If yes, why wasn't this author paid for his review? If not, why are you considering an exception? This won't be a one-time incident: if you write again to those reviewers, they will want to be paid again, because you set a precedent. If they tell colleagues they were paid, you will be short of reviewers because everyone will want to be paid and will refuse to review otherwise.

Point 1. is the only right thing to do; if you want to be malicious, maybe also 2., but that's still a bit unprofessional to me.

In general, the author's and reviewer's role (even for the same person) should be separated. Being a reviewer is mostly voluntary, and it's just agreed/expected in the community to act as a reviewer from time to time. You cannot force anyone to do it. But you are obliged (as an editor of a publisher's journal) to consider for publication papers that you receive. Just get over this, and maybe consider avoiding working with this person as a reviewer or other such roles in the future. But don't dismiss him as an author. Don't be vengeful.

Any potential reviewer with appropriate expertise who declines to review for a journal is effectively saying "my time is worth more than that". This individual is trying to find an equitable middle ground. Sure, paid reviewership very uncommon and totally outside the norms of academic review, but it's completely indefensible to take vengeance on this person because he feels that journals don't appropriately value reviewers' time.

Anything other than accepting the submission and reviewing it like any other is petty, discriminative, and potentially damaging for your journal's reputation. I know I would have serious reservations about submitting or subscribing to a journal that rejected papers irrespective of their content and based solely on personal vendettas with the author (option #5).

Peer review isn't a mechanism to get back at people.