I've pushed rounds/cases out with dowels before with relative ease. This one will not budge at all. The rim of the case is wedged into the very edge of the barrel. I was thinking of building a tube threaded for the barrel muzzle and then use a threaded plunger on the other end to use oil or grease to force the cartridge out with hydraulic pressure.
Good idea, dumb idea, bad idea? Any tips are welcomed.

Wood dowel, cleaning rod, tap out. If it takes more effort than that to get it out, you must have had a hell of a time getting it in.  What caliber?  Is it a reloaded round?  Have you ever used a chamber brush on this rifle?

Beat me too it on the wood dowel and cleaning rod.  Only time I have done something similar was slugging a Mosin, just put a length of wood dowel down the barrel to contact the bullet and drop a cleaning rod long enough to give you several inches of clearance so your not swinging something right near the crown of your muzzle and tap it out. Personally I just used a 2' piece of pine 2x4" so it didn't much matter if I hit anything I didn't mean too. If that doesn't work, impressed and I suppose your hydraulic idea would work...just use oil? If you have to resort to that, you should make a video and post it for us.

Dowel is not going to work, I broke two trying. The rim was damaged before hand and is what is holding it into the chamber. Wasn't hard getting it in there, sure will be hard to get out. I don't have a great way to make a video, but I'll see what I can do.

Can you take still pictures of the problem?

What caliber is stuck?

300 Blackout.

The long story is this: I have a pile of few 'bad' reloads. Stuff I don't want to shoot but keep for testing or will disassemble later goes in this pile. I load 300 Black for my AR with a 203 gr bullet at 1.97" COL. Testing this load in my new Handi left rough chamber marks on the brass but also marks on the projectile (Hi-Tek coated lead). I seated a cartridge from my 'bad' pile shorter (1.94" IIRC) to see if that would have any marks from the chamber. The problem is this round had a damaged rim, which is why it was in the bad pile. I didn't think anything of it, but it must have had just enough material outside the normal rim size that got pressed between the case and chamber wall when I closed the action.
I was just smart enough to use electrical tape on the end of my wooden dowel that went down the barrel but not smart enough to tape up the end I was hammering. In the past, this has worked fine. I split both dowels at the hammer end. So, at least there is no wood wedged between the bullet and bore.
The extractor pulled a chunk out of the rim. I did try to pry it out through the extractor notch a little bit but didn't really like the risk of damaging anything, including possibly the primer, so I didn't mess around too much with that method.
In hindsight, if I shot the thing it would have been fine, even at 1.94" and would have been a heck of a lot easier to get out. Now I have no idea what the bullet looks like or how far back in the case it might be, so that is definitely out.

Not to criticize, but I would suggest not using your "bad pile" for anything at all.

Fill bore with motor oil.
Dowel close to bore size.
Insert dowel into bore.
Smack dowel with hammer.

Cut down rod that just fits into barrel into 2-3 inch sections put into barrel then hit it, takes the flexibility out of the rod this way

My idea would be a brass rod that is just under the diameter of the inside the barrel with a countersink in it, push it down the barrel so the tip of the bullet seats in the countersink, then smack it with a 2x4. In all likelihood the bullet will be pushed back into the casing but this should do it. I also would pull the upper receiver off of the rifle before doing this.

The pointed tip of a rifle round can crush swaging the bullet into the bore, when displaced by a squib rod. BTDT

I would go straight to the hydraulic displacement method.  Use grease not oil.  Oil will push out around the projectile, and threads under pressure.

I'd try freezing the barrel overnight...
Next day, try the dowel/rod tap out when you pull it out of the freezer...
If it doesn't budge, let it warm back to room temp and try again.  The difference in dimension changes between brass and barrel steel might just free things up enough to break it loose.