Usually unis won't let you do more than one language from scratch at the same time, particularly with ancient languages (although they may let you start from scratch in one the first year, then pick up the other in second year).
Usually intermediate level in the languages presupposes either GCSE or A-level in the language - more than you can pick up in a summer. With (ancient) language work it is absolutely
essential you get the best grounding in the basics you can. Much better to start in the beginners course and thoroughly learn the material with guided tuition than try and scrape yourself into the intermediate course and end up with major gaps in your knowledge which prevent you making good translations.
So I would a) recommend only doing one and b) if you have not done a formal
qualification in either before, start from beginners level.
As for which one to take, it depends on your particular interests. Obviously for the philosophy side, Greek is very useful for an classical philosophy to be able to read Plato/Aristotle/etc in the originals. That said, if you wanted to continue to postgraduate level Latin would probably also be desirable as many of the surviving editions may well be in (medieval) Latin. Latin is also generally useful if you wanted to work on medieval or early modern philosophy in the original as well I expect.
Outside of that I'd observe Latin might be slightly easier to manage as a beginner to either language simply because it uses the same script as we're used to (the Latin script, obviously!), and doesn't have (as) much of a system of accentuation, unlike Greek (which is both in a different script, albeit somewhat familiar - but with false friends! - and has an accent system which you would need to learn properly). Easier to focus on the grammar when you don't have to put in extra effort just trying to read the text initially! While granted you get used to it reasonably quickly anyway, it is just one less thing to think about while you're getting used to thinking about grammar in a more explicit way than you probably have been