I'm camping at twin harbors state park now but am less than satisfied.

What are some spots to camp by a river to fish that I can have a fire and possibly shoot?

Somewhere in Alaska.

There's a lake up in the Wonder Mountain Wilderness that has some great fishing.

There are no trails into or within the wilderness though so you'll need to be able to navigate through the forest without getting lost or dead.  The hike in is about eight hours over very steep terrain, so you'll have to make some decisions about what to bring.
These are pics from the first time I went in:








Here's a few from the time I went solo:

No fire allowed.
Firearms might be illegal.

Why post it up when it does not meet the OP's stated requirements?  

http://www.fs.usda.gov/recarea/olympic/recreation/recarea/?recid=78493
Fires prohibited above 3,500 feet elevation within Wilderness.
Fires prohibited at Wonder ponds and all ponds in the Wilderness. Stoves only.

http://www.fs.fed.us/recreation/programs/trails/welcome.shtml
If you are planning on visiting a designated Wilderness Area, the Regional Forester or Forest Supervisor has the option to implement a special local order which additionally prohibits the mere possession of a firearm within that Wilderness Area.  
Although this prohibition is not common, you should contact the Forest Supervisor's office to find out whether such a restriction has been imposed.

Wasn't aware.
Might be, but aren't.

You should be able to camp, shoot, and have fires on the National Forest (North) side of the Carbon River (South side = Mt Ranier National Park -> more restrictive)...



I've gone hiking in the area & seen numerous roadside 'shooting areas' on the road up to the trailhead....



I also presume there are fish in said river...

Spent the last couple days on the pend O'Reil river in three north east corner of the state. Meets all the requirements for you, though all we caught were bass.

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Spent the weekend SE of Randle at the forest service campground st mile 11 of FS23. There's places to shoot nog too far away.