This room is almost always visited immediately after opening the chests in Compass Room in order to collect the small key underneath the pot in the top of the room, even if not proceeding onward (unless the big key and at least one small key are already on hand).
The fastest way through the first part of the room is to dash (or walk) along the edge of the pit to avoid triggering the first set of star switches. Be sure to release all d-pad inputs during the dash, otherwise the dash will be interrupted when touching the edge of the pit.
You can optionally hookshot to the pot after dashing to save time, but make sure that you hold up after doing so to ensure that you are flush with the pot.
The above screenshot shows the vertical pixel range for which this works -- between coordinates 0E7A and 0E7F. If walking directly south out of the portal (optimal), this is usually a 4-frame window, but this can be increased to 6 frames if you walk diagonally instead.
Using the hookshot to grab the key while walking downward doesn't save time over an optimal walk, though it may be slightly easier to time. Ideally you'll want to lift the pot from the left side, so that you can walk to the right while waiting to pick up the key.
As with the first part of the room, a dash along the edge of the second pit can avoid the star switches. If the dash is started flush against the wall (or if you nudge around the pot) you can keydash through the door as well.
Without the boots, you'll have to adjust your movement to prevent the conveyor from pushing you into the pit.
Because the conveyor belt pushes Link downward (even while dashing), and because you can nudge around the pot, the vertical coordinate range for the second dash is different than the first. Link must be between vertical coordinates 0E71 and 0E7B when starting the dash in order to make it through successfully.
While starting at 0E78 or lower will avoid a nudge around the pot, this is undesirable because the conveyor belt will push Link out of position for the keydash. Because of this, you'll actually always want to nudge around the pot in order to ensure proper pixel alignment. Ideally you'll start the dash as low as possible (0E77 is optimal), but as long as you nudge the pot you'll be setup for the keydash.
If the second bari is electrified, you can use a sword beam or the hookshot to dispatch it before dashing through, to avoid having your dash interrupted.