Looks like you're going to have to work on it cocked. Not the end of the world if you remain careful and don't get frustrated. Leave the port closed. If you discharge opened, it will block the exiting air and might mess up your gun. I would get something like fairly stiff foam, say from a swimming pool noodle (great stuff for all sorts of things) and cut a thick piece to wedge in behind the trigger. Enough so that an accidental pull will not fire the gun. This will give you a bit more safety when working on it. Of course, you can do the same to the safety lever, wedging it on.
I would not put too much oil down the bore in trying to dislodge your obstruction. Unlike firearms, airguns don't do well with a lot of oil in the spring chamber. The dowel rod is maybe your best bet, although finding one hard enough for a .177 might be a neat trick. Make sure it is oak since poplar that small will be next to useless, being too light. If you have a .177 cleaning rod, you might be able to get away with a jag tip that's been filed flat. The last thing you want to do is to have the rod go off-center and wedge in amongst the other stuff. (This from muzzle loader experience.) Best thing is to go slow.
Good luck. And no more skewers. Save them for the victory barbeque when you get it cleared.