Ah, apologies. The Super Edge gives you the benefits of the Edge without actually giving the character the Edge itself. Block give the +1 Parry, etc. The problem with Trademark Weapon is that you can't give it the Weapon, you're right. Let me see what I can do with that. In the meantime...
The normal work-around, use a Permanent Adjustment to add the Trademark Edge to the character and select the proper weapon, won't work here for the same reason. There's no weapon to apply everything to when a Super Power like Attack, Melee (or attack Ranged) is based in a device. Things that should affect melee weapons don't affect it because there's no actual weapon on the character sheet.
Let's create a solution. What kind of weapon are we talking about? Melee? Go into the Editor, and create a new file. Name the file whatever your weapon is called.
Go to the Hand Weapon tab, and create a New (Copy) of whatever the weapon is.
Blank out the entries for Weapon Die, Special Damage, Attack Bonus, Damage Bonus, Parry Bonus, Armor Piercing, and Weapon Reach (unless you want to keep that last one).
Change the name of it to your weapon, and change that question mark in the Unique ID to as much of your weapons name as you can get away with (don't change the wp prefix!).
Also set the Uniqueness drop-down to Unique (unless you have more than one?).
Unclick the Treat as Standard Gear check box.
Open up the Source tab and click either Super Powers Companion or whatever Source you are using for the game.
Save the record, and then click on Test Now! in the upper left hand corner.
What we're after is a blank template of your weapon that does only Strength damage.
Go back to your character, and the weapon should be available on the Armory tab, Hand Weapons. Go ahead and get it, and equip it.
It should now be available to pick up on the Trademark Weapon list.
Now, we can get away with this for Melee weapons because Attack Melee affects all hand weapons. Ranged weapons don't work that way, so this won't work the same way with them. For Ranged Weapons, I'll have to come up with something else.
I'll see if a better solution is available when I go through the next set up core updates.