Sudden Charge allows you to use it with any special speeds you may have, including a climb speed.
Climbing requires two free hands. Weapons generally also require free hands.
Does this mean Combat Climber is required to be able to Sudden Charge while using a climb speed, and that it is impossible to do so with a 2h weapon, or a shield, etc?
You can use any attacks that don't require your hands, Combat Climber not necessary.
Sudden Charge also doesn't require you to make an attack, so you can use it without making an attack (though there's no mechanical benefit to doing so).
But if you'd like to use a 1 handed weapon, you'll need Combat Climber. And two-handed weapon users are SOL.