I have a 3G module in the post so I can't help yet but maybe once it arrives we can collaborate. I did however buy the sim5216 because there appears to be a lot more AT command manuals related to HTTP access. I'm not sure what the difference the two modules is though. The sim800-900 had very similar AT commands, so with any luck the 5216 and 5320 will be similar too. Have you tried using the SIM5216 at command manual?
I also can't remember exactly why, but there was a reason I couldn't use SIM900 modules with my gmail account. I think it was to do with port access on googles end not allowing the type of connection the GSM module makes. Not sure if the SIM5320 has a way around that