I recommend recruiting an Electrike before the battle. Electrike's Lightning Rod ability absorbs all of Zapdos' electric attacks. You can also use a Geodude due to its high defense, typing, and strong STAB Rock Throw to use.
By now, your Pokemon should be level 17-18ish. I recommend training in Mt. Thunder itself instead of Silent Chasm since the Pokemon there are the highest level available. Electrike comes at a competent level anyways, and at Level 16 Geodude gets to abuse Magnitude to KO the strong electrics Electrike can't hurt. Obviously, don't use it on Zapdos.
You may also want plenty of Gravelerocks. Treecko can use them with Quick Attack for a very spammy strategy while the other members wallop Zapdos. Before long, you'll win.
Edit: Just to clarify, Zapdos will still try to use regular attacks on you, but at least with Electrike you won't have to worry about getting a Thundershock rained down on you. Agility is still a problem, though.