Hi eureka9, I don't think this has to do with the APN settings. I believe this is an issue with AT&T based on my research with my dad's Xperia ZL.
Based on what I have read, AT&T has a list of approved devices (or IMEI, not sure which) that can use the LTE network (basically the phones you can buy from AT&T). Because Sony's new devices (XZ, XZL, XZU, etc.) are not offered by AT&T, the best you can get is HSPA.
The way to get around this is to have your line associated with an approved LTE device (e.g.: SGS3) then just take the SIM and put it in your ZU. I came off of a Galaxy Note 2 but I didn't not call AT&T to "activate" or anything, just put the SIM in. My ZU connects with LTE (when there's reception of course but that's a whole other issue). I can do a screenshot and post if you want (just need to know how to do that, got my ZU just yesterday and loving it).
With my dad's Xperia ZL, he was coming off a Moto Atrix 2 which has no LTE capabilities. Calling AT&T and activating the Xperia ZL didn't help as he still only got HSPA. I did a test and put in my SIM from my Galaxy Note 2 (this was several months back BTW) and voila, LTE connection.
As mentioned, I did some digging when working with my dad's XZL and besides having a SIM already activated with a LTE phone, some AT&T stores may be nice enough to put a "dummy" IMEI from a LTE phone (like a demo unit or something). If you Google "AT&T Xperia ZL LTE" (or something like that), you should find some posts on other forums that talk about this method (but YMMV depending on the store). I haven't done this myself but I'm planning on switching to T-Mobile anyways.
Hope this helps.