This works, but probably not the way that you want it to.
Let's start by looking at the relevant portion of Double Slice:
Requirements You are wielding two melee weapons, each in a different hand.
You lash out at your foe with both weapons. Make two
Strikes, one with each of your two melee weapons, each using your
current multiple attack penalty...
You are holding a melee weapon that doesn't require two hands in one hand and you are wearing a gauntlet on the other hand. Now let's look at what qualifies as Wielding:
Some abilities require you to wield an item, typically a weapon.
You’re wielding an item any time you’re holding it in the number of
hands needed to use it effectively. When wielding an item, you’re not
just carrying it around — you’re ready to use it...
You are holding the one-handed melee weapon with one hand, and the gauntlet is being worn on your other hand. You have met the requirements for wielding both the one-handed melee weapon and the gauntlet, and thus have met the requirements for Double Slice.
The gauntlet has the Free-Hand trait:
This weapon doesn’t take up your hand, usually because it is built
into your armor. A free-hand weapon can’t be Disarmed. You can use the
hand covered by your free-hand weapon to wield other items, perform
manipulate actions, and so on. You can’t attack with a free-hand
weapon if you’re wielding anything in that hand or otherwise using
that hand. When you’re not wielding anything and not otherwise using
the hand, you can use abilities that require you to have a hand free
as well as those that require you to be wielding a weapon in that
hand. Each of your hands can have only one free-hand weapon on it.
This allows you to use your gauntlet hand for actions that require a free hand, such as Grapple, Snagging Strike, or Combat Grab. However, performing a Double Slice after a Grapple or Combat Grab will end the effects of the previous action, due to this section of the Free-Hand rule:
You can’t attack with a free-hand weapon if you’re wielding anything in that hand or otherwise using that hand.
Grapple states:
You attempt to grab an opponent with your free hand.
(Emphasis mine)
Similarly, the wording for Combat Grab is:
If the Strike hits, you grab the target using your free hand.
(Emphasis mine)
These highlighted sections clearly indicate that your free hand is now being used or occupied, which prevents you from being able to attack using that hand. You may have met the requirements for Double Slice, but the specific rule that your Free-Hand weapon cannot attack while being used is overriding the more general rule that you can Strike with each of your weapons (in accordance with Pathfinder's Specific Beats General game convention).
In order to be able to use Double Slice, you would have to Release whatever you're holding with your free hand as a free action, ending the effect of the Grapple or Combat Grab.
Your Grapple or Combat Grab would end, but not your Snagging Strike
The Snagging Strike action has no such limitations within its wording, however, which means that you can, by RAW, use it and then Double Slice while keeping its effects. I'd like to imagine that a fighter grabs, tugs, and releases a creature while using this action, using several small grabs that don't rely on keeping a continuously hold. While the creature is within your reach, you can keep it off-balance for the duration.
Best Answer
When thrown, the weapon is ranged.
From the Thrown trait:
When you Strike with a thrown weapon, you choose whether it will act as a melee weapon (by attacking with it normally) or a ranged one (by throwing it). In this case, Precise Strike would grant additional precision damage on melee attacks with the Combat Grapnel, but not its thrown attacks (though you could use Flying Blade to get around this restriction within the first range increment).
As far as I'm aware, Pathfinder 2e lacks the distinction like the one mentioned in 4e D&D. Some actions that require melee weapons will use language like the Swashbuckler, ex. "a Strike with a melee weapon or unarmed attack," while others (like the Magus's Spellstrike) specify you must "make a melee Strike with a weapon or unarmed attack." These appear to be interchangeable.
In a similar case, the cantrip Produce Flame has a melee and ranged use. You select whether the cantrip will act as a melee or ranged spell attack when you cast it.