I will give you two methods by which you can own the guy. One is a simple method, the other will take a while altho since it is based on the team you provided. I will also strongly suggest that you get yourself a Milotic. When I fought in the tower against random trainers and palmer it was my MVP. It didn't die even once.
For the simple method you will have to raise one of the most annoying Pokemon to find and catch, Milotic. Once you have it give it the moves Ice Beam, Recover, Toxic, Surf/Hydro Pump. Give it any item you like tho when I used mine I gave it zoom lens to increase the accuracy of both Toxic and Hydro Pump. With Milotic, you will basically just destroy all of the random trainers even if they have grass and electric type moves since it is bulky enough to take at least one hit from those type of moves and the AI rarely use moves like Swords Dance.
When you go up against Palmer for the silver print, lead out with Milotic. He will then either use Dragonite or Rhyperior then use the other as his second Pokemon. His last one will always be his own Milotic. Take out Dragonite and Rhyperior with the appropriate move. Once you face his Milotic, use toxic and just spam recover until it dies. Or you can switch out and use another Pokemon, specifically your Raichu to take it out. On this challenge it is important that you use a special water type move on Rhyperior because it has really high defense and hp stats while having an abysmal special defense. You can use your Floatzel on this challenge if you don't wanna bother with Milotic, however you might have a slightly difficult time as Milotic has better stats. So for the silver challenge use your Floatzel/Milotic (if you do decide to train one), Raichu and Gallade. Your Gallade will simply be the back up while your Floatzel should take out Dragonite or Rhyperior. Do not use Raichu unless you are facing his Milotic. Side note, if you decide to keep on Floatzel, then give it a focus sash so it will be guaranteed to take out one of palmer's Pokemon. This strategy however, could fail if your water type of choice fails to take down one either Rhyperior or Dragonite and not severely damage the other.
For the gold challenge it is rather tricky. I suggest using Staraptor as a lead with focus sash, Houndoom and Gallade. If palmer sends out Regigigas, then take it down with Close Combat. The focus sash should allow you to survive whatever he dishes out so you will be guaranteed to take it down. The next Pokemon you would face should be Cresselia. At this point use your Brave Bird. Staraptor will go down but will deal a good amount of damage. Send in Houndoom and finish off Cresselia with Dark Pulse until it dies. For Heatran just break the focus sash and let Houndoom die. Send in Gallade to take it down with a single Close Combat.
If palmer decides to lead with Cresselia, then switch out Staraptor and send in Houndoom then just spam Dark Pulse until Cresselia dies. If Houndoom dies before Cresselia does then send out Staraptor again and finish it of. A single return should be enough to take it down. By this point you have two Pokemon who have fighting type moves and are faster than both Regigigas and Heatran, so just use Close Combat to take them both down. Alternatively, you can use Gallade as your second Pokemon and finish off Cresselia with Night Slash however I do not recommend this as Night Slash isn't stab for Gallade and Cresselia has a rather high defense.
Note that all of this is just based on prediction, as in my experience, palmer and all of the other frontier brains have a habit of not using their Pokemon in a set order. They will always lead out with the one who has a super effective move against your lead Pokemon. The key to winning in the battle frontier for me has to always been to predict the Pokemon my opponent will use, then switch out accordingly and use a different Pokemon who can take down theirs.