How do you think your manager would react if you booked a 1-1 to discuss your role, and you talked though your reasons for wanting to leave? If you can go in with a list of things you can complete within 4 weeks, would they agree to it? If it is possible to negotiate, then I'd say that is the best solution.
Some years back, I was offered a great role but needed to start in 4 weeks time. My notice period was 3 months, so in my resignation letter, I was very positive about my current company, and praised my then manager. I then stated the new role was a terrific opportunity for me to progress my career, therefore I wanted to agree an earlier leaving date than the standard 3 months, and for my final date to be 'on or around xx/xx/xxxx' which was in 4 weeks time.
Can't say that my then manager was happy about this - in fact he was absolutely livid because there was a recruitment freeze on, so no hope of replacing me. But the company was a multi-national, so references were very brief, factual statements prepared by HR, and I just kept my head down and worked exceptionally long hours during those 4 weeks to make sure that I completely documented everything I did, and that the handover was as smooth as possible.
I don't know if he took a leaf out of my book, as he never spoke to be again after I left, but I heard though former colleagues that when he handed in his own notice the following year, he only worked 4 weeks notice as well :-)