It's due 4/15, no extensions. Some other things like SEP IRA's can be paid when your extended return is due, but not T.I.R.A.'s.
Another thought though - If you reduce your tax withholdings today, you will have more net check to save by 4/15, then you can ramp up your withholdings after the refund comes in. Go to HR, fill out a new W-4, and write EXEMPT. They will withhold $0, but you have to make sure you make up for it later. Can you get any type of loan to bridge the gap? HELOC, family, 401K, employer loan, cash out some taxable investments??
You could also temporarily stop all 401K contributions, then catch up from May-December. This could have a very big impact, and since you will be writing EXEMPT on your W-4 your taxes won't increase either.
Also, based on the numbers you're in the 15% tax bracket. I think a T.I.R.A. is still worth it, but some would argue ROTH IRA would be better.