YOU CANNOT GET YOUR VISA IN SEOUL.
Don't go. Japan either. I'm assuming you're not a permanent resident of one of those two countries, of course. Maybe you can do it in Mongolia, but I don't know the logistics.
Go to Hong Kong. There's a reason that everyone does it. What is expensive about it? Last time I went I took a round trip to Shenzhen for about 1400 RMB, and took the bus/subway to HK. Very convenient. When I went it was a couple thousand more to fly directly
to HK, and I had time to kill, so...
I'm curious about Macau as an option as travel there is quite convenient too, but I never hear of people doing it.
By the way, when you request the issuance for your Z visa, you select the location to have it issued from, and I hear you may have problems if you don't pick the place you end up going.