Sorry to burst your bubble about timing, but this is not like hanging a window unit and plugging it in. You are not likely to get this done in a week nor will it be inexpensive.
The first thing you need to understand is that a two AC Airstream needs 50 AMP electrical service. Unless your trailer used to have a second AC, it only has a 30 AMP service. So you will need to upgrade your electrical service to 50 AMPS. Look around the Forum with the search tool and you will find some DIY hacks that people have done to their electrical systems to provide the necessary amount of power for a second AC without actually upgrading to a 50 AMP service. I am not sufficiently schooled in electrical engineering to know if those setups are really safe or meet code.
Once you have adequate power to the trailer, adding a second AC unit to the roof is actually the (relatively) easy part.
One thing you can do quickly is to cover the windows with water heater insulation. That stuff looks like thin bubble wrap with aluminum foil on both sides. You would be amazed at how much heat you can keep out if you put that stuff in your skylights, in between the windows and screens and on the inside of the panorama windows. It obviously makes for a very dark trailer, but the dogs might not mind.
One important thing to keep in mind whether you have one AC unit or two, you need to have someone keep an eye on the trailer to let you know if the power goes out and you lose the AC while you are gone. This happens more often that you might realize in commercial campgrounds and it would be a tragedy if this happened and no one knew to rescue your dogs!