Ok, I just got my Cisco WAP121… and everything is running super smooth. When you fire up the AP the first time, you are presented with a config wizard; I simply entered VLAN 40 when it asks for the wireless VLAN. Didn't have to touch anything else. And now everything works perfectly. This makes me positive the D-Link DAP-1353 is either broken, bugged, or doesn't comply to the networking standards.
At least the time spent on this "project" wasn't entirely wasted. I've honed my VLAN'ing skills, and learned a couple of new tricks :)
AP only needs to be vlan capable when you want to run different SSIDs on different vlans
I figured I'd need VLAN to separate the web interface from the guests, so I'd be able to config/snmp without having to access their network directly. Could this be done differently, even without VLANs?