If you can figure out the naming convention for the EXST / EXSEAT (e.g., just call the operating carrier and ask?), I'd just book the award online using that (e.g., LASTNAME, EXST). You'll have to pay all the taxes that might otherwise be waived, and you may not get the refund that you're not really interested in anyway, but I think the chances that you run into a problem at checkin/boarding are very small. You may not be able to do OLCI without a passport for EXST, so obviously plan a little extra time for counter check-in.

And frankly, even if the naming convention isn't exactly correct, agents deal with name mis-spellings all the time.