I can't speak with authority for every airport, but generally airside lounges don't get direct mail/courier deliveries. You can only have something shipped there if you address it somewhere landside then an airline employee (or fellow traveller) drops it at the lounge. So that probably won't work. You could definitely have something delivered at the Hyatt but they might charge you to hold it if you aren't a customer. Probably best to phone them up.
Personally, I'd look for a landside luggage storage place and phone them and ask them to accept the delivery for you, since holding items is basically their business.
Finally, there is a US Post Office on the airport grounds at DFW (but not in the terminal), you could always use that as a last resort.