The hardest part to skipping the Fishing Rod is not skipping the item itself (since it's simply a matter of doing the Day 1 & 2 skip), but rather beating the game without it. In a normal playthrough, the only use for the Rod is to use it to catch a reekfish and obtain it's scent, which allows you to find the path up Snowpeak. But, when the Map Glitch allowed for early Snowpeak, and the Zora's Armor became skippable, the Fishing Rod became a required item again for obtaining waterbombs so to enter Lakebed Temple early.
There are several ways to enter Lakebed Temple early without the Zora Armor, mainly Pillar Clip.
In order to skip the fishing rod without rupee diving into Lakebed Temple or completing Goron Mines, it is required to get the Zora Armor. The only way to do this without the bow and arrow is to use bombs to defeat King Bulblin during the Telma escort. This is made far easier by using the item menu to set up the right position to throw the bombs, as King Bulblin's hitbox is incredibly small.
After reaching Kakariko Village, you need to do an LJA over the first King Bulblin trigger to avoid activating it.
In the second room of the dungeon, you will encounter a number of ledges that would normally require bombs with the assistance of either arrows or the boomerang to knock down the stalactites. The fishing rod would normally be required to fish out some bomb fish from a digging hole to do this, however the below methods allow you to skip doing this.
From the end of the bridge, pull out the Gale Boomerang and target the collumn towards the left or right side of the door and do a long jump. From here, climb onto the fallen stone slab, and target near the torches on either side of the door and use another long jump to reach the next plaform.
Get on the end of the bridge and use the Gale Boomerang to lure a Helmasaur towards the edge of the above platform. Once he is near enough to the edge for you to target him, do so, and pull out you sword. Back up a bit and do a long jump off of the Helmasaur. If he is too close to the edge, you will have to wait for him to turn around otherwise your jumpslash will bounce off of his armor and the recoil will prevent you from landing on the platform. From here, turn into a wolf and climb to the crest of the fallen boulder. Align yourself, and jump to grab the ledge of the next platform.
Although this method can be done on either version of the game, it typically used on the Wii Version, as it is slower than simply using Long Jumps on the GameCube.