Sorry for the somewhat necro, but I completed a "not-speedrun" speedrun earlier today, trying to test out a few different possible options if someone wanted to go for a truly optimized playthrough, ending up with about a 4 hour time or slightly less. Some differences from the route above:
-I didn't bother with a GiantWorm, I bred my Armorpede directly with one of the Orcs (you can recruit one from the "Beavern" battle and get Biino after beating Oasis) to save time. I bred that HornBeet with the Armorpede from the Arena to get better stats, and this lasted me for the entire run.
-For most of Pirate, I used a team of an Orc, the BigEye "CurseLamp", and a HaloSlime, as the HaloSlime knew three very important skills of HealMore, Revive(!!!), and Sleep. Given how tanky it was, this made it very easy to go through all the boss battles. I switched out the Orc for Squiz when I recruited it.
-I recruited a Wyvern from Oasis to breed with the Shadow to get a MadGoose, and I recruited a Pixy to breed with "Squiz" to get a Merman, all for a WhipBird that was ready to go before "BombCrag". The only Metaly grinding I did was to get the MadGoose and Merman to sufficient levels to learn TwinHits and such, meaning I was able to get a WhipBird with both breath attacks, HealAll, Vacuum, TwinHits, PaniDance, and Revive, which I used for the rest of the run
-For my last party member, I bred the other Orc with a Dumbira to get a Grizzly with MetalCut and RainSlash for some easier battling, and I bred that with the Golem ("BombCrag") to get a Roboster with very strong stats. I bred the Grizzly, despite it having better attack growth, because its other stats were awful; the Roboster also had strong attack while being quite tanky too, and TwinHits+QuadHits is the optimal way to take care of most bosses in Sky and Limbo.
-I made sure to get as many TinyMedals as possible, which as it turned out wasn't terribly necessary; since the OrcaCape and WarRing sell for a ton of gold too, I didn't actually need to go overboard, but it definitely makes things easier.
-To compensate for not doing any breeding or grinding beyond the bare minimum (I defeated three Metabbles in the cave and used a Repellent because it's a massive time sink actually killing them), I used my excessive amounts of gold to buy WorldDews, which I spammed during battles so my monsters could focus on dealing as much damage as possible. Defense is for cowards. The hardest section in the run besides Kid's Class is probably Metabble cave simply because it's hard to deal enough damage quickly. Ice World is quite easy if you're dealing enough damage (and by using three of the best monsters in the game you certainly are), and Sky and Limbo are cakewalks after that.
I'm pretty sure with better routing and RNG (I spent 10 minutes trying to find a Wyvern of a different gender until just going with one to match my Shadow, and I think I could have gone for WorldDews earlier), you could get this under 3 hours. Random encounters can be avoided with Repellents probably after you level up your Roboster to 15 or so, which should happen without too much effort, excluding the encounter or two in Metabble cave, where you really want to get critical hits early. Otherwise, it's just not worth the experience, as you get a hefty amount from boss battles and you don't want to waste MP. Being sure to recruit ArmyAnt on your first pass through Oasis also helps, as well as finding things like the female Pixy quickly (you want a Pixy to pass TwinHits to your WhipBird later). The Roboster may be an unnecessary indulgence, but I don't know if the Grizzly would survive.