We also just had one bathroom with a "sewer smell".

In our case, when the water was first turned on in the sink, there would be a blast of the foul odor. First I thought it was the water itself, but then I decided to catch the water in a pot - no smell then. But when I poured it down the drain, up came the smell. Aha, perhap ssome bacteria, etc down the drain. So I poured a bit of bleach down the drain and that took care of it.

Hope this helps solve your problem.

(BTW, if that wasn't the cause, my next step was to pour some bleach down the sink & tub overflow drains.)

Your thought of the vent stack is probably correct. City sewer folks probably can't help you, though, since the problem is on your side of the sewer connection. You'll need to cal a plumber/handyman if it's not a DIY job for you.
I would snake the vent stack from the roof. You'll need about 50' or so of snake to do this. Most houses have more than one vent, depending on the locations of the drains, I'd snake all of them.

BTW, I doubt it's the vent line, but to check, you might try to isolate the wall cavity that it's in from the drain below. Maybe you could plug around any opening with rags and stick a piece of one inch tubing up in there and smell the other end with your nose. Crazy, but hey, it just might work.

I once dealt with a bathroom that had a vent line that came apart since one joint wasn't glued. In that case it caused a ceiling leak when it rained. Do you have any signs of moisture in that wall?

Are you sure the shower actually has a trap? You'd hear bubbling if the gas was really flowing through a trap filled with water. A DIY shower by previous owner may have been installed incorrectly.

We have an older house (1920) with a basement bath added in the 60s or 70s. The shower and toilet are not vented. If we remodel this bath, we need to vent. Ergo, no remodel, as it is such a freaking hassle...