The way I would do it:
Select the blocking face and delete ONLY THE FACE, not its vertices or lines.
You will have to make a ring of vertices that match where the barrel enters that face. The ring will have to be parallel to the face. I’ll explain how to do that in a minute.
If the ring and the face are two separate objects, join them (in object mode select both ring and face object, and ctrl-J to join them.
I am going to assume that the barrel extends along one axis (say the X axis). This will make the face parallel to the other axes (in this case the Y and Z axes).
Select the vertices of the face. Shift-S Snap Cursor -> Selection. The 3D cursor will snap to the center of the face.
In the lower menu bar Pivot control, set the pivot to 3D cursor. The pivot point will snap to the 3D cursor.
Carefully select one vertex of the ring. If you miss and the pivot point moves, you’ll have to go back and repeat the last two steps.
Ctrl-L to autoselect all vertexes of the ring.
S-key, X-key, Num-Zero-key, Enter
(assuming barrel is along X axis, otherwise replace X-key command with one for the correct axis)
The ring should snap to the center of the face.
In the lower menu bar Pivot control, set the pivot to Median Point.
Now comes the tedious part. You will have to make lots of triangular faces between the corners of the face and the vertexes of the ring in order to fill in the face. The end result will be a face with a barrel shaped hole in the center.
Making the ring
If the barrel is of constant width, this is easy. Select the barrel. Enter edit mode. From either the front or rear part of barrel, select the outer ring of vertexes. Shift-D, Enter to duplicate them. P-key Selected to move them into a new object. Tab to enter edit mode. Click on the new ring to select and merge it with the face as per instructions above.
If barrel is of varying width, this will be a pain.
Select barrel in object mode. Shift-D, Enter to duplicate the barrel. Select the duplicate and enter edit mode.
A-key until entire barrel is selected. Use number keys 7, 1, or 3 until you see the barrel and the face (in separate object not being edited) edge on. Now you will have to K-key Knife (Exact) and cut along the line defined by the face.
After cut, select the old vertexes and delete them. Enter object mode and merge the new ring as per instructions above.
Let me know if you have questions.