Most of those block-size lots downtown are rental parking, and except for those that stay open to collect money from people visiting entertainment venues like BOK Center, they tend to be empty at night, and some may be closed. Tulsa tends to be pretty quiet at night, streets and lots empty until the morning rush. But I suspect that if you dropped the chain at the gate of one of those empty lots, and drove in for the night, you might be towed before morning.
On-street parking might be empty late at night, but I don't know hours for time restrictions and parking meter operation. These tend to change from block to block.
If you really want to stay at the Hyatt Regency, I suggest calling their concierge service about parking. They most likely have someplace where valet service parks visitors, and can almost always do something for you at a price.
Another thing you might do would be to take the trailer out to the Expo Center where parking is usually free (unless there is a special event like a ball game or a race) and tends to be empty at night. I see trailers stored there often, belonging to exhibitors at Expo events. Drop the trailer in the area where you find others parked, then drive back downtown where your truck might fit in the Hyatt's parking garage. When it is time to leave you can go back out to the Expo center to see if your trailer is still there.
Or you might try to rent a RV space in the Expo Center RV park, to store your trailer.
