This is a fascinating question, and to help you develop your analysis further you will find a link below to the character analysis section of enotes. Here are my ideas however. Firstly, let us remember that Shakespearian fools are very different from Shakespearian clowns. Fools in Shakespeare's plays seem to have a dual purpose of actually making some quite serious comments about characters and what is happening under the guise of acting as a "fool." However, clowns seem to be more figures of fun, who are there for amusement purposes alone.
If we examine the character of Puck in this light, he does appear to be more of a clown than a fool. Note how he introduces himself in Act II scene 1:
I am that merry wanderer of the night.
I jest to Oberon and make him smile
When I a fat and bean-fed horse beguile,
Nieghing in likeness of a filly-foal;
And sometimes lurk I in a gossip's bowl,
In very likeness of a roasted crab,
And when she drinks, against her lips I bob
And on her withered dewlap pour the ale.
It is clear that Puck is a mischievous character, who delights in playing pranks on mortals. It is interesting that normally plays make Puck a kind of sidekick to Oberon, and some suggest that he deliberately confuses the Athenian lovers to act as a kind of force of chaos in the play. This would point towards him being more of a clown than a fool. Yet at the same time, let us not forget that Puck is given one of the most important lines in the play:
What fools these mortals be!
This would indicate that he is actually more of a fool, as this line sums up one of the key themes of the play: how we can act so stupidly when under the influence of love. Thus I would argue he is a clown, but with some of the qualities of a fool.