There is no recourse. Roth conversions are irrevocable, made so by the Tax Cuts and Jobs Act of 2017, and you are obligated to pay the taxes. You can request a payment plan with the IRS or you could take a distribution from the Roth IRA to pay the taxes, but if you are under age 59½ you may be subject to an early-distribution penalty on some or all of the distribution.
"I ticked the box in the form that said 'Convert the entire account'."
This is not a bank error. It would likely be fraudulent for the bank to "reverse" the conversion and issue a corrected Form 1099-R that indicates that no such distribution ever occurred from the traditional IRA.
This might not be as bad as it seems because the $585,000 would eventually be taxable anyway, albeit likely at a lower tax rate. What can now remain in the Roth IRA will grow tax free instead of tax deferred, so there will be some savings on future taxes from that that will somewhat offset the taxation at the higher tax rate.
I'm not sure why you would even have been attempting a "backdoor" Roth since the majority of any amount converted would be taxable since you had such a large balance in traditional IRAs.
