It shows deep thoughtfulness that you are considering the character of the Fool in this way, for this is the kind of questioning that has kept scholars discussing and analyzing this (and all of Shakespeare's plays!) through the centuries.

However, if you are interested in a definitive answer, then I'm afraid you'll simply have to rely on the literal evidence of the text.  And, as far as characters in his plays, Shakespeare was clear, obvious and direct.  If a character appears onstage and is visible to all, then that character holds conversation and is included in the action.  And this is true for the Fool.

If Shakespeare, on the other hand, meant a character to be an apparition or visible only to one character, then that is made very obvious in the text.  In Macbeth, the ghost of Banquo appears at Macbeth's banquet, but it is unarguably clear from the words that the other thanes and Lady Macbeth speak that Macbeth is the only one who sees the ghost.

In Hamlet, the ghost of his father is visible and audible to him but not Gertrude, when he visits her bedroom in Act III, and Shakespeare makes sure the audience knows this by the words that Hamlet and Gertrude speak to each other.  And since Shakespeare does not have any character in King Lear remark about the King's imaginary Fool, then we must assume that he is as real as any other character onstage.

I'm afraid that there's simply no evidence for a playwright in the Renaissance writing anything that is meant to be "ambiguous" or that contains "subtext" of any kind.  These are modern inventions and, while it is tempting to apply our modern methods of analyses to Shakespeare's plays (and can provide very interesting points of departure when staging the plays), there is absolutely no evidence to support any supposition that Shakespeare had any such "hidden" agendas in mind.

The upshot here is that if Shakespeare had meant for the Fool to be a figment of Lear's imagination he would have made this fact quite obvious in the language that the other characters speak.  However, this is also a very interesting concept to utilize when considering staging the play, and I encourage you to stage a scene for your classmates to test it out!