That is a shallow trap. Certainly not of legal dimensions for a sanitary
drain. The workmanship of the plumbing overall isn't encouraging.

I'm trying to figure out if there's a way to somehow intentionally
empty the trap so I can see if the smell comes without having to run
the kitchen sink.

Plumb in a trap with rubber Fernco couplers. This lets you open it up for
testing and blowout.

But while I can understand how pressure differences caused when the
kitchen sink runs could suck water out of the p-trap, I don't
understand:

a) how this connects to the condensate line, as I thought those were
separate from any line the kitchen sink would connect to, or

Your condensate drain is connected to the house sanitary drains, right?

You may also have a venting problem on your house drains.