If you're not doing a lot of towing, not towing at max gross weight, and not going up a lot of hills or over mountains, then a trip or two towing without the trans cooler is probably ok. You might want to do a drain/refill before and after the trip though. I wouldn't flush just for this, just get some of the fluid swapped out in a drain/refill and it'll probably be fine. Probably
If you're doing to do a LOT of towing, then a GOOD trans cooler is a great idea. Almost any shop or dealer ought to be able to tell you if you have the tow package or a trans cooler already installed, get a service call done at a good shop or dealer and have them check to see if one is installed while they're doing the oil change or whatever. Cheap trans coolers use rubber hoses to/from the cooler, an OEM kit ought to use metal lines and will last longer, leak less, etc. A cheap trans cooler might be ok to save money if you're not going to keep the car forever, but if you use a cheaper trans cooler then you will need to inspect the trans cooler lines periodically and replace them if there is any sign of leaks or getting old, etc.
When I bought my 2010, everyone I knew who had an Ody said that the trans was the weak point in the whole car and the trans cooler was the true engineering fix, so I got the whole OEM towing package and OEM hitch when I bought the car new. I haven't had any trans problems in 200k miles.
I'm not 100% sold on the need for a PS cooler for occasional towing. I've had PS problems with my car since it was new, with high steering effort required in parking lots whenever the rpm is low. Increasing rpm immediately makes the steering feel normal. The PS pump has been replaced twice under warranty and once on my own expense, and it still does the same thing. My totally unfounded and unjustified suspicion is that the PS cooler might be contributing to a loss of PS pump pressure at low RPM since the dealer basically has no fix other than replacing the PS pump every year or two, and it doesn't seem to be a widespread issue for people who have had the PS pump recall or TSB accomplished like I have (twice).
