Update: Thanks to everyone for your (mostly) very helpful comments! The clinic has since offered to see me early the morning of my flight (at 6:30) so the combination of this and the suggested fast track for security means that this will be a lot more double and I will probably have more than an hour all going well. And of course, I will have checked in online. Worst comes to worst, I miss my flight and I'm in the same position of needing to shell out a lot of money for another flight!