[RPG] Would a free hand from gauntlet axe let me attack two handed with a bastard sword

No

You are considered to be attacking with your second hand on the weapon.

You are holding the weapon in your other hand, thus it violates the "with anything in it" clause.

Depending on the DM's interpretation, this may be a possible strategy...

Two-weapon fighting relies on the DM's definition of the slippery verb wield:

If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. (PH 160).

And that's because what wield means in game terms is unclear, and whether wield means worn or used in an otherwise empty hand like most other light weapons is certainly made no clearer by the spiked gauntlet description. Thus...

1. If the DM rules that wield means worn, then two-weapon fighting with the spiked gauntlet and a two-handed weapon is a valid strategy. In such a campaign, a creature is always wielding its spiked gauntlet while it's worn. That means a creature's full attack looks like you describe in your question: take the full attack action to make an attack with the falchion, take a free action to remove one hand from the falchion, make an off-hand attack with the spiked gauntlet, and take a free action to return that hand to the falchion.

2. If the DM the rules that wield means used in an otherwise empty hand like most other light weapons, then this isn't a valid strategy. In this case, if a creature's wielding the spiked gauntlet, he's not wielding the two-handed weapon because one hand wields the spiked gauntlet, and if he's wielding the two-handed weapon, he's not wielding the spiked gauntlet because both hands wield the two-handed weapon. He can't ever simultaneously wield both the spiked gauntlet and the two-handed weapon, therefore failing to meet the wield requirements of two-weapon fighting.

...But try to avoid some arguments by using armor spikes/spiked armor

While the DM must rule whether a spiked gauntlet is an appropriate off-hand weapon while also employing a two-handed weapon, a creature wanting to use two-weapon fighting while wielding a two-handed weapon could employ armor spikes/spiked armor, which says specifically

You can also make a regular melee attack (or off-hand attack) with the spikes, and they count as a light weapon in this case. (PH 124).

But, despite that text, the DM may determine that nothing's changed the armor spikes/spiked armor from a light weapon, and to wield this light weapon still requires the weapon to occupy 1 otherwise free hand. This ruling returns the two-handed weapon wielder to the same spot he was in when he attempted to use the spiked gauntlet.

In other words, ask the DM, and hope for the best.