First I'll comment on the bolded individuals where needed, then I'll get into how I interpreted your ideas and comment on that.
Rapidash
It seems like Rapidash is highly dependant on either Swagger or Flash Fire where it otherwise under-performs. Setups that boosts only one member is often times not desirable, but if you do decide to leave that open as an available option you should make sure that it can do a sufficient job even without the boost.
Braviary
The main reason the gang-up happens here is because your opponent may not find your other members sent out really threatening, which tends to happen sometimes if you have two Pranksters out that aren't attackers thereby leaving you with one attacker. Aerodactyl does do a quick Tailwind and isn't always expected coming from it, nor is its use of Wide Guard which takes care of incoming Rock Slides etc. At the same time it's not expected because it takes up moves that could be something else instead. Its Tailwind is defeated by things faster which includes scarfs, rain teams, and chlorophyll-sleep where Talonflame's +1 priority Tailwind may have more successful executions. Consider Wide Lens + Stone Edge for a rock attack on Aerodactyl if you were going to give it Rock Slide. This is because if the opposing team has a Wide Guard user out (which should be recognizable) and you've sent out Aerodactyl, chances are that they'll think a Rock Slide may be coming in. Aerodactyl should probably have one more attack and Protect is obvious. I'd lean towards Talonflame but since you want an uncommon Tailwind user Aerodactyl is an option. It could also knock out Noivern at a Jolly nature before the opposing Noivern can use its Tailwind.
Klefki
1/3rd rather than 1/2 in Doubles and Triples. Sticking to only one screen instead of two is also something to consider, which opens up options for other things like Quick Guard, Thunder Wave, Fake Out. Reflect is probably a good screen to stick with.
Seaking
Consider leaning towards consistency over situational-benefits. You'll want to have options that can be utilized across most battles rather than waiting for a specific encounter. Swampert hits very well with Avalanche, Earthquake, and Hammer Arm. You'll want Protect on it, so if you want to give it Wide Guard you'd have to give something up. You probably wouldn't use Earthquake with your team composition though outside of Protect/Braviary, if you know your adjacent partner is going to faint prior to the Earthquake, or weighed sacrifice. Rock Slide is also an option if you're having difficulty with things like Charizard Y and Talonflame but I'd stick with Avalanche and throw in Life Orb. Waterfall can be another option but only if you feel that you'd need it which I don't think you would.
Trick Room difficulty
Imprison for reliability over Taunt when there are multiple Trick Room users out, though it takes up two slots. Gardevoir is a good option for this. For opponents that bring out two Trick Room users you could alternatively have two Taunt users.
Wanting to use only your favorite Pokemon can sometimes be difficult to work around, but this is what I like to do as well for some of my teams. This only gets tougher when you have a fixed selection of the Pokemon you want to use, and you're trying to formulate an idea from it when the other way around would be a lot better to work with. What you've got here are ideas that usually don't complement each other too well. Let's go over again on these ideas and the ones your team relies on to work effectively:
- Safeguard/Swagger: Self-boosting
- Tailwind: Team-boosting (Reliant)
- Screens: Team-boosting
- Flash Fire/Lightning Rod: Baited-boosting (Reliant)
Remember that even if there are three slots, your opponent will be utilizing their three slots as well. Executing Tailwind takes up one slot, so you're left with 2 to deal with the opponent. Safeguard/Swagger is a 2-turn here, so that's another slot and depends on surviving. Even though your team may not rely on Safeguard/Swagger most of your team relies on the Tailwind to avoid fainting so chances are that if you ever choose to use the Safeguard/Swagger it would be used on the same turn as Tailwind. Tailwind is usually accompanied by unboosted heavy hitters or as an extra for already fast teams that may need the extra boost (something to consider?). Your two reliants can work together but I've already mentioned the issue with relying on baited-boosts rather than having it just on the side. The screens option is more flexible. When reworking your team, think about complementary ideas and play out the scenarios cognitively.