How to check, if my ASUS Nexus 7 LTE really is LTE version?
The issue is most likely caused by a wrong APN configuration. Though customer care told you their card is "self configurable", this may well be relative: many ROMs (especially custom ROMs like cyanogenmod) come with a built-in list of APNs, which are matched against the provider as specified by the SIM card, and used correspondingly.

While those matches usually are fine, there might be exceptions – e.g. when the provider made changes to his configuration after the ROM was released. Or did not provide all details: APNs might need different parameters for e.g. 3G than for LTE. As you've got a second Android device where the card is working fine,


put the SIM into that device
go to Settings (usually network → telephony or similarly; different ROMs might place it slightly different)
tap the APN and select to adjust it
do not adjust anything, just write down its parameters


In a second step,


put the same SIM to your N7
go to the same Settings page
compare your values and, if something differs, adjust it


I bet your device then will connect fine. If not (or if there are no differences), it's probably a hardware issue, and you need to send your N7 for service.