The Enbrel travel pack should be good enough with the ice pack. Also, since the airlines are really strict now...if you have a written Rx of the Enbrel, perhaps bring that with you too. I've known a few folks that had problems bringing insulin and syringes on flights...the security guys were asking for written documentation! They were suspicious of what was in the syringes. The medication will hold well at that temp in the cooler, it will remain stable as long as you do not leave it out at room temp for too long.
Hope your travels are smooth and safe!
How has the Enbrel been working for you?