What is the dramatic significance of the climax being represented in the storm in King Lear?

I have to disagree with one central element of your
question. The storm, which occurs in Act III of this tremendous tragedy, does not mark
the climax of the play. This of course happens in the final act, Act V, with the French
invasion and the deaths of various characters and Lear's final descent into
insanity.

So, the storm then serves to mark the rising
action of the plot, and of course, it is very symbolic in terms of how it mirrors the
action in the play. As Lear roams around the blasted heath in Act III, the storm echoes
his own inner turmoil and his obvious madness that is only worsening through what he is
experiencing. It acts as a kind of pathetic fallacy where the natural elements are made
to express what is going on internally within Lear. However, let us also think slightly
wider than this. If we think of the focus of the play, which is on the humbling of the
arrogant Lear, the storm is a symbol of the tremendous strength of nature which makes
Lear recognise his own frailty and forces him into a state of reluctant humility. Note
what Lear says in Act III scene 2 as he addresses the
storm:

I tax
not you, you elements, with unkindness;

I never gave you
kingdom, called you children.

You owe me no subscription.
Then let fall

Your horrible pleasure. Here I stand your
slave,

A poor, infirm, weak, and despised old
man.

Note how the storm
forces Lear to confront his true state in front of the brutal, elemental strength of
nature. He stands alone, with all pretensions stripped away, and recognises his own
weakness.

Some critics also argue that the storm could be
considered to bear some reflection on the turbulent state of anarchy that has descended
upon Britain now that Lear has unwisely relinquished his power to the wrong
daughters.

Therefore, although we cannot link the storm to
the climax of the play, it is clearly a very important symbol that marks the play's
rising action.