Rentals in October, staying past lease date, etc.

Have they offered you a renewal lease yet? I wouldn't offer more money at the start of a negotiation. Let them ask if they feel it's warranted.

They offered to renew the lease on attractive terms. The problem is that I have a studio but need a 1-bdrm. Their 1-bdrms are priced in line with the similar size apartments but in nicer buildings in the neighborhood and they won't budge. So I informed them that will not renew the lease. However, I am facing a very busy September at work and it will be very difficult to move, so need an extension.

ask for a month to month extension?

I wouldn't offer an increase, let them ask you. I'd put the difference between summer and winter, peak-to-valley, at around 10%.

the dirty little secret here is that if you need to stay an extra month, there is almost nothing they can do to stop you. if you send in your rent check for the 13th month a few days early, they'll almost certainly cash it. usually they won't even file any kind of legal notice for at least one month after your lease has expired. they want to avoid the monstrous hassle that is new york housing court as much as you do.

What do you mean by attractive terms? Is the renewal cheaper?
If so, they probably would accept your old rent for a 1-month extension. They really should, because if you don't send in the renewal, pay the old amount and send them a letter letting them know you will vacate by the end of the month, there's not much they could do to stop you.

My rent was reduced 19% in 2009. The offer is for no increase, which is about 6% below the going rents for in the same line in my building. I would love to take the offer but got engaged and we need a bigger place.

I already spoke to mgmt office, they agreed to let me stay another month for the same rent.