So I've been playing around with this, trying to break it, and the most acute problem seems to be in a hypothetical effect that looks something like
"You constantly shed bright light with a radius of 5 feet x your proficiency bonus."
That is: You don't roll anything, nobody else is rolling anything against you, there is no discrete point to say "You should roll a proficiency die here". I don't think there's any real choice but to use a constant value here. I suppose it's
possible to say you roll for it once per day, or whenever you finish a short rest, or whatever, but that feels forced.
For that constant value, the obvious option is to go with the standard proficiency bonus for a character of your level. But, like I said, that feels kind of disappointing to me: you still
have a proficiency bonus, it's just hiding. So I'm currently playing with the idea of instead using the minimum value of your proficiency die (
i.e., normally, 1). This doesn't seem too punishing to me because, unless I'm forgetting about something big, these effects are really rare. And it much better sells the flavor that, as a fool, you really are just bad at things whenever chance isn't involved.
As for defining when to use the proficiency die, I like @
Blue's thinking a lot. I hadn't quite considered the problem in terms of actions vs. passive effects. The problem is passive ability scores. Technically they're, well, passive, in that you don't use an action to activate them, and so would use the static bonus. But conceptually, a lot of the time passive scores do seem to represent repeated or abstracted activity, as has been discussed on the numerous other threads about them. And if I go with my static-bonus-is-1 idea, then applying that to all your passive scores would be rather punishing. So my current draft looks more like a generalized version of what @
Charlaquin suggested: the defining quality is
whether or not somebody is rolling a die.
Roll of the Dice
As a fool, you live by the maxim "Better lucky than good" -- even if you don't realize it yourself. You do not have a normal proficiency bonus. Whenever you would add your proficiency bonus to a roll, instead you roll your proficiency die, a d4, and add the result.
If you would add your proficiency bonus to a DC, such as a spell saving throw DC or a passive ability score, you roll your proficiency die and add it to the DC each time a creature rolls against it. (If the DM wishes to roll in secret, he or she may instead roll your proficiency die and subtract the result from the creature's roll.)
If you would apply your proficiency bonus to an effect other than a roll or DC, use the minimum value of your proficiency die (normally 1).
If a feature would let you double your proficiency bonus, roll your proficiency die twice and use the total. If a feature would halve your proficiency bonus, roll once and divide by two, rounding down.
Your proficiency die increases in size as you gain levels, as shown in the Proficiency Die column of the Fool table. If you are a multiclassed character, your proficiency die is based on your total character level (and you do not get a proficiency bonus from any other classes).
(Oh, and Blue, I'm definitely not just referring players to the table in the DMG with this. If the fool doesn't have a "Proficiency Die" column instead of "Proficiency Bonus" right there in the class table, well then, what's the point?

)
Also trying to decide whether I can get away with calling it a "chance die" rather than a "proficiency die". I do still have to call your various skills and weapons and whatnot "proficiencies", so that might be stretching things too far.