It's not necessarily your fault if you can't get it done in time - you were thrown in to a position you weren't familiar with - though I stand by the fact that you should tell your manager you are in need of help with it - whether it's internal or not. It will show that you are able to handle a project even when it has potential to fail, which is a real world scenario - it's just as important of a skill as being able to get stuff done.
If you feel that BigCo, and companies of that size are really not your thing, then you need to do what will make you happy. It's a fact that you will at times have to do work you will not like, that's just life, but if you are sure that the larger company environment is not for you, which is fine - it's not for everyone, then you need to make a decision... if you feel that your lack of experience will hamper your progress though, then it might be in your best interest to wait it out and gain that experience at BigCo which hopefully is a big name that will look good on the resume. This is a big decision, make sure you read mine and everyone elses advice and really contemplate what you will be doing - even if it's early in your career.