It is possible to cross the flame circle plaforms without activating the eye switches by landing at the corners of the platforms using either a sidehop and jumpslash, or a angled jump.
If you are swordless by either having a NG+ item on B or if you just did not get the Kokiri sword, you will need to defeat the Lizalfos with deku sticks. However, it takes three jumpslashes to defeat them. So here you need to perform the Broken Deku Stick glitch by jumpslashing a Lizalfos up close. It is easiest if you stun one of them with a deku nut and then jumpslash.