Advice for pushing back start date?

Just got my first job offer in consulting. I'm graduating on May 21st. In the interview they really emphasized how it's going to be >40 hours a week sometimes with late nights and weekend hours. When I was offered the job on the phone yesterday, the recruiter told me the start date on the offer letter is June 9th, but that's not a hard date and it can be moved. Would asking to start August 4th (the first Monday of August) be pushing it too much? I'm graduating with my masters and I have a basically full time internship now that I kind of feel like I need the break, especially since the full-time job can be late nights and weekends sometimes. I wasn't really expecting to get an offer this soon tbh and ideally wanted to start work in August. But is asking for August too late? I'm a little nervous, this will be my first full time job ever. What about mid to late July? It's a relatively small firm (~25 people). Thanks for the help!