Kakashi feigns disinterest?? I think he's really not interested at all in fighting Gai from the beginning. Whereas Gai wants so much for Kakashi's attention and recognition. Something like Naruto and Sasuke.

But intense rivalry can turn sour, just like Orochimaru and Jiriaya.. Same with Naruto and Sasuke, when the more talented one realises the "idiot" is getting better and surpasses him.

Kakashi's the smart one by not caring at all about rivalries with his cool and 'so what' attitude.