Go take a look at the Blackberry Priv threads here, from when it was first released. The first release was the unlocked version; AT&T was getting a version, but it wasn't out for awhile at first. And everyone was having this very problem, and the solution is exactly the same.
It turns out that AT&T's systems won't simply turn on LTE just because. They are set to recognize IMEIs from AT&T-branded phones, period. If it sees that IMEI, it will turn on LTE.
If it doesn't see that IMEI and your SIM isn't set up in their system as having been on an LTE device, no go. So the answer is to pair the SIM up with an AT&T-branded LTE phone first (take it to the AT&T store if you have to), let their system see the combination and record it, and you're good to go. Your SIM is now set up in their system as working fine with (known) LTE, so now when it sees your SIM the system will enable LTE on your account.
So just plug your SIM into your unlocked non-AT&T branded phone, and voila.
And if your phone has AT&T's LTE bands in it, you're good to go.
This is simply not true. I received a brand-new nano-SIM for my LG V10 (H901). Inserted it, applied the APN settings from ATT's site and 4G service only, no LTE. I had previously checked the bands in my area before purchasing the phone, they were B4 (on my old LG G3, which had a microSIM). The H901 should support B4. Using the APN settings above, I now have LTE service. This SIM has never been in another phone, so your theory doesn't hold water.