Oof, that stagelist for Big House 3 is heartbreaking. I've grown very wary of the bottom row, and it's been a gnawing at me that we're looking at very few different stage types. I mean, looking at the bottom row you have
4 Triangle stages (BF, YS, DL, FoD*)
2 single tier platform stages (PS2, SV)
1 Flat stage (FD)
*FoD changes its design all the time, but it's more considered a triangle stage than not.
But you also have: six flat main platforms, and one stage with angled edges (YS).
You also have three static stages(FD, BF, PS2), and four changing stages (YS, DL, SV, FoD). Of these changing stages, only Smashville and Fountain really have a huge effect on gameplay at all times. Randall can be a guy, but otherwise he doesn't overtly affect the match at all times.
The rest of this post is just my own personal opinion.
For any tournament I run in the future, the starters would likely be:
Lylat Cruise(Angled edges, unique platform structure, but respectively fairly flat, with Melee Battlefield edges.)
Final Destination. (Wide, Flat)
Battlefield. (Traditional 3-tiered stage)
Pokemon Stadium 2. (A brilliantly flat design. 2-tiered, 2 floating platforms. Wide stage.)
Wario Ware. (Small blast zones, walls up the sides, 3-tiered, four floating platforms. Narrow stage)
Fountain of Dreams. (Moving stage. Asymmetrical. Psuedo 3-tiered, platforms rising and lowering means adapting strategies. Larger blastzones, narrow stage)
Skyloft. (Wide Stage, Asymmetrical. Small blast zones. Unique terrain can help keep the match flowing.)
And my picks for the Counterpicks would be
Dracula's Castle (Crazy platform shenanigans, wide horizontal blastzones, wide stage, lowish ceiling, walls that go on forever)
Green Hill Zone (Narrow stage, narrow blast zones, and a unique platform setup. Flat with Walls.)
Norfair. (Two tiered, flat, but unique side platforms and a large, central floating platform which can change up the game when.
Yoshi's Story (Three tiered, Randall, angled edges.)
Yoshi's Island (Angled main platform, angling floating platform.)
Dreamland 64. (Triangle, wind, large stage)
SSE: Jungle (three tiered stage, inverse trapezoid. Walls and narrow blast zones)
I'd probably run with three bans. I know it's a controversial stage list, but you can generally cover most stage types within these stages, with a couple noteworthy stages thrown in. Stages I didn't let in, like Metal Cavern, generally aren't chosen because their job is done better by other stages(in this case, Skyloft). They're both asymmetric stages with a higher right side over the left, but Metal Cavern is hella small. Personally, I'd really like to have added that 3-tiered, asymmetric Summit I see floating around on a lot of Wii's, as well as Frigate Orpheon(which would replace Yoshi's Island).
I really like Lylat Cruise now that it doesn't tilt, and am at least very glad it is a Counterpick at Big House 3.