Pros... if you are very very lucky, and injury doesn't strike you again, then your enthusiasm will take you far.
Cons... in all likelihood, you'll get injured again. Put back your fitness regime, and potentially become disheartened. You could end up undoing a lot of the massive strides you've made these last couple of months... all for the sake of being a bit impatient.
If you play it by the book, you should cancel the 5K you've entered, because there are hundreds of 5K races you can enter every single weekend (www.parkrun.org.uk) But if I was you, I'd continue with the C25K and then when Sep 7 arrives... if it is really really important to you, then give the 5K a go at a very very steady pace, but then have a few days' rest, and then resume the C25K where you left off, to allow your body to recover and adapt to all this training.
I've only been a member of one club... I was always a "lone" runner, enjoying my own company when running.
Now I'm still a lone runner for 80% of the time but run with my club perhaps once a week... maybe slightly less often than that.
I love the camaraderie - developed between massively different characters. Some gregarious, some positively awkward, but all seem to be decent people and somehow get on really very well. You might be surprised how social it can be - because you can just join in a run with a group, and barely say a word, and it doesn't much matter because many people are concentrating on their running.... but when it does come to small talk, it is so easy because the environment is so focussed on running, you know for sure that there is always at least one subject that you can talk about, and know that you have something in common with the group.