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.