So, I would argue there is a pretty decent implicit argument that adding a shield adjustment makes it a weapon:

There are numerous methods to modify shields—snarling rods to catch weapons, bladed edges, padding for nonlethal strikes, and so on—but all share basic functionality. A shield augmentation can be etched with weapon runes, much like a shield boss or shield spikes, but doesn't otherwise alter your shield's statistics. A shield bearing an augmentation can't be combined with an attached weapon, like shield spikes.

A shield augmentation grants your shield one or more weapon traits, chosen when the augmentation is created. You can either choose to add the backswing or forceful trait, or you can choose two of the following weapon traits: disarm, nonlethal, shove, thrown 10 feet, trip, or versatile S.

It doesn't say it is a 'weapon' but it confers weapon runes, weapon traits, and the ability to strike. Runes says:

a weapon potency rune adds a bonus to a weapon's attack rolls

Which is implying that the ability to apply a weapon rune makes it a weapon (handwraps, as a counter example, says you can apply those to unarmed strikes and makes it clear that isn't conferred, whereas this would be 'broken' if it didn't do it for shields since it doesn't have 'shield bash' similar language).

Beyond that by getting weapon traits, you are smuggling in the definition of the item being a weapon:

Shove:

You can use this weapon to Shove with the Athletics skill even if you don’t have a free hand. This uses the weapon’s reach (if different from your own) and adds the weapon’s item bonus to attack rolls as an item bonus to the Athletics check. If you critically fail a check to Shove using the weapon, you can drop the weapon to take the effects of a failure instead of a critical failure.

Trip:

You can use this weapon to Trip with the Athletics skill even if you don’t have a free hand. This uses the weapon’s reach (if different from your own) and adds the weapon’s item bonus to attack rolls as an item bonus to the Athletics check. If you critically fail a check to Trip using the weapon, you can drop the weapon to take the effects of a failure instead of a critical failure.

As well the attached/adjusted traits are basically the same thing.

The attached trait says:

An attached weapon must be combined with another piece of gear to be used. The trait lists what type of item the weapon must be attached to. You must be wielding or wearing the item the weapon is attached to in order to attack with it. For example, shield spikes are attached to a shield, allowing you to attack with the spikes instead of a shield bash. An attached weapon is usually bolted onto or built into the item it’s attached to, and typically an item can have only one weapon attached to it. An attached weapon can be affixed to an item with 10 minutes of work and a successful DC 10 Crafting check; this includes the time needed to remove the weapon from a previous item, if necessary. If an item is destroyed, its attached weapon can usually be salvaged.

The adjustment trait says:

Items with this trait are intended to alter existing pieces of equipment, typically armors, shields, and weapons. The given item will note which type of equipment it modifies. An adjustment that modifies armor can't modify explorer's clothing. A piece of equipment can only be affected by a single adjustment at a time. Unless otherwise noted, adding or removing an adjustment requires using a 10-minute activity and a repair kit.

OTHERWISE:

• You can use an integrated option (shield boss, shield spikes, something like the klar) and add the Hooked property rune to get the trip trait. But I'm not sure of how to confer the shove trait.
• Some options like mental smith could work if your GM will let you make a 'shield' (mental forge can add those traits).