"Signature weapon" isn't clearly defined. There's one section that says:
Distinctive weapons
• antique handgun (2-harm close reload loud valuable)
• hidden knives (2-harm hand infinite)
• ornate dagger (2-harm hand valuable)
• ornate sword (3-harm hand valuable)
• scalpels (3-harm intimate hi-tech)
None of these are more effective than their common counterparts,
so they exist solely to give your characters character.
Operators get a “signature weapon.” It might be one of these
distinctive weapons, or it might be some other weapon made
distinctive and personal.
So the rules as written lean toward your MC's interpretation that an operator's signature weapon shouldn't have any special functionality like harpooning someone and dragging them around.
On the other hand, the MC is supposed to be a fan of your character, and a harpoon gun is cool. The tag set (2-harm reload messy
drag-your-victim) don't seem severely out of balance with the other options.
To me, it seems like the letter of the law supports your MC, while the spirit of the game supports you. At this point, you're into social contract territory.
If I was your MC, I'd say yes, but I'm not your MC.