Sell your tent and get a nice TarpTent Moment. It's a single wall tent that weighs 28 oz. with 2 stakes. The Moment's ventilation is excellent. I love mine.
Not saying you won't get some condensation once in a while, (esp. in the UK) but that it will be minimal unless you camp beside a stream or river.
Your tent will be just fine. No need to worry. You will have the time of your life.
I did the GR10 in one hit myself for charity a few years ago and took a Marmot Eos. Condenstation is not generally a problem in June - obviously you will get some wet weather and possibly some legendary thunder storms when it can rain very hard. If you are going to do the whole thing in one hit you will have great days and bad days - warm cosy nights and a few damp (or even cold) nights. Nothing too severe is likley though. Just be sensible where you pitch - the usual stuff.
Will be down the eastern end in June myself - so will keep an eye out for you!