Um, I would strongly suggest against it, to be frank... It is possible to do as such, but the creativity and overall quality will suffer. As you said, it's not a simple mix-work or just foley, it's the entire soundtrack (minus the music, I guess judging by your post). When I did a theatrical mix for 2-3 minute promos for upcoming movies a while ago it took 2 days just for the mix and nothing else (it was a pretty heavy patch-work to dig into). If you're inexperienced, you will need more time though to get to know the material and figure it out. Especially dialogue editing can be very time consuming if it's not perfectly recorded.
How long that's reasonable differs from project to project, but assuming this is a pretty straight forward drama or similar, with not too much true design to take care of, I'd normally need about 5 days and up for 3 minutes of everything except music, provided the sync dialogue and music isn't a mess. As you said you had only done this for six months so far, I'd say you would do yourself a favour demanding at least 7 days, which leaves you some time to think things over as you go, something that's extremely valuable when one hasn't had the time to work up one's routines yet! Or just going on routine is plain boring ;-)
Historically, I've done 3-minute shorts that had very heavy designs taking at least 2 weeks or more to do, which has included heavy interconnecting layering, several hundred tracks, and busses, all with different perspective and selective acoustical placements, but I've also had gigs where I had to invent entirely new previously unheard sounds and soundscapes, mainly for surreal projects though, and such things can sometimes (if permitted to) take A LOT of time to build from scratch, especially when it's actually used as an active part to tell a story, and you get to work closely with the director.
Also, a good reason to not wanting to do everything in too short time - hearing fatigue. What you think you hear when about to call it a day is rarely what you'll hear the morning after, and sometimes (not very rare either, to that), things that felt like a very good idea at the time when focusing completely on one part, can prove pretty horrendous when you get perspective on it, meaning there are several reasons you want several days to do stuff and get them right! The brain works best when getting to process new ideas :-)
If I may ask, what type of project is it?