I agree with your accountant, for two reasons. First, the IRS will always re-bill you for something. You probably owe interest along with the underpayment penalty, and the interest will be re-calculated based on exactly when you pay. The exact under-payment penalty is also only final when calculated by the IRS, and for reasons known only to them, the tax accountant or the software almost never gets it exactly right. So you will almost certainly get another bill anyway with some kind of adjustment.
Second, if this is your first time owing a penalty to the IRS, once you get the bill, you can apply for a one-time waiver of the penalty. Any interest you owe can’t be waived, but it will be recalculated if the penalty is lowered or removed.
There’s not really a separate penalty for not paying the penalty. Any unpaid amounts after April 15 (or July 15 this year), will be charged interest at a variable rate that is currently around 3% APR. If you wait to get a bill for the penalty, the interest will be slightly higher that if you paid it now, but that would only be a few dollars at the current interest rate.
