What's the current objective?
To take on the Elite Four. I doubt the current team is up to it, but as in Red the plan is to grind the E4. But the Red team, while not perfect, was more balanced (by types/attacks/levels) than this one.
Shuckie - Rock/Bug. Level 27
Strength
Constrict
Wrap
Encore
Only normal attacks, and two out of three are useless.
Brian - Pigeot, Flying/Normal. Level 50
Fly
Quick Attack
Whirlwind
Wing Attack
One of the best on the team.
Hoot Hoot - Flying/Normal. Level 31
Peck
Hypnosis
Reflect
Take Down
Totally outclassed by Brian, in both level and moveset. Weakest link on the team imo (Shuckie with his high defense/sp.defense is more useful than this).
Gator - Water. Level 57
Surf
Whirlpool
Cut
Rage
Basically carries the whole team, although shout-outs must go to Brian as well.
Espeon - Psychic. Level 38
Bite
Tail Whip
Sand-Attack
Growl
Our Psychic type knows no Psychic moves. Although the Dark type Bite does make use of Espeon's high special attack, and flinching is useful. But only one attack move!
Katie - Dragonair. Dragon type. Level 30
Waterfall
Thunder Wave
Dragon Rage
Twister
She can actually do some work, if she survives long enough. But level much too low (Elite Four average level ~43, Lance ~48). Dragonite does not evolve until 55, fyi.
Now let's look at the Elite Four in G/S/C:
Will (Psychic team): Xatu, Exeggcutor, Slowbro, Jynx, another Xatu
Not too bad a matchup type-wise. Luckily, Will's Exeggutor doesn't know any Grass moves (other than Leech Seed), so the heavy hitter Gator won't get taken out too early. But every one of his Pokemon know Psychic, which will hit all but Espeon for good damage. Espeon will be useful with Bite as his first attack, so he should be put in front for A-spamming, although I don't think Espeon can OHKO all of them.
Koga (Poison team): Ariados, Venemoth, Forretress, Muk, Crowbat
This one will be more trouble than people anticipate, I think. 3/5 know Toxic, so expect several of our team to be badly poisoned in the following battles. Ariados knows Giga Drain. Forretress will take forever for anyone but Gator to kill, and he has Explosion as a lovely parting gift. None of our team's attacks are super effective against any of Koga's Pokemon except the bugs.
Bruno (Fighting team): Hitmontop, Hitmonlee, Hitmonchan, Onix, Machamp
A favorable type matchup trainer. But Hitmonchan's elemental punches could give Gator, the birds, and Katie trouble. This time around his Onix is no joke, with Rock Slide (bad for birds & Shuckie) and Earthquake. Machamp knows Rock Slide as well, and Cross Chop destroys things.
Karen (Dark team): Umbreon, Vileplume, Murkrow, Gengar, Houndoom
Houndoom can be devastating with Crunch and Flamethrower. None of our team's attacks are super effective against any of her Pokemon except Vileplume (Flying) and Gengar (Bite, but Espeon will probably have fainted to the Dark attacks by now, if not in an earlier battle).
Lance (Dragon): Gyrados, Dragonite, Charizard, Aerodactyl, Dragonite, Dragonite
He will slaughter anyone left alive at this point.
All six of his Pokemon know Hyper Beam. In addition, his Dragonites know Blizzard, Fire Blast, and Thunder, respectively. Charizard & Aerodactyl have all strong attack moves, and Gyrados has 3/4 attack moves. Lance packs multiple Full Restores as well. Suicide by PC will be less painful than this battle.
Edit: Katie, Brian and Shuckie were just deposited lol. Although I think Katie was withdrawn again.