Yes, you should definitely put it in writing. But you don't need to be formal about it. Just send him a letter or an email, setting out what you expect to do and how much you will charge. Then - most important - ask him to write back to say that he agrees. The whole thing can be competely informal. And it won't antagonise the client in any way. On the contrary, you are showing that you are business-like.
You asked: "Would there be any way to enforce or prove that we met, for the timeframe that we did." Don't worry about it. Providing your charges are reasonable and you don't try to take advantage of the client, he is highly unlikely to dispute the time you spent on the work. If he does, that would indicate a breakdown in trust, in which case you might decide that it's better to write the client off.
Also, I wouldn't invoice in advance. Do the work to the best of your ability, and submit an invoice. If you don't get paid within a reasonable period (say, 30 days), send him a reminder. If that doesn't work, there are further steps you can take. But you are a long way from having to worry about that just now.
Finally, a couple of articles that might help:
Freelance contracts: A cautionary tale
Freelance contracts: Covering the essentials