If a borrower expects a higher tax bill next year (construction assessment vs completed assessment), and they want to deposit additional funds to cover it...is that allowed? I know the bank cannot require additional dollars (other than the 2 month cushion), but if it is borrower initiated, could we do that? Thanks!

Yes, customers do it all the time.

I know the bank cannot require additional dollars (other than the 2 month cushion),

That is a false statement when dealing with new construction. You should be anticipating the increase and build it into your initial escrow analysis as to not create a payment shock to the consumer the following year:

1024.17(c) (7) Servicer estimates of disbursement amounts. In cases of unassessed new construction, the servicer may base an estimate on the assessment of comparable residential property in the market area.

Thanks for the clarification!

While Randy is right, I'm going to push back slightly on the "build it into your initial escrow analysis" comment he made. You need to be careful of when the higher property tax will actually be paid out. For example:
I'm building a new home now (April 2016).
The construction is anticipated to be completed in the fall of 2016.
My home should be assessed at the new new/higher value when construction is completed.
If my home is accessed at the higher rate, that goes into effect in 2017 in my state.
2017 real estate taxes are paid in arrears in my state. Therefore, the higher real estate tax won't be actually paid out until 2018.

HUD encourages servicers to provide borrowers with a notice explaining that they anticipate disbursements from the escrow account to increase substantially after the first year. The borrowers may make voluntary overpayments in anticipation of the increase. This disclosure is not required, but suggested. Here's what RESPA says about the Payment Shock Agreement:

After an initial or annual escrow analysis has been performed, the servicer and the borrower may enter into a voluntary agreement for the forthcoming escrow accounting year for the borrower to deposit funds into the escrow account for that year greater than the limits established under paragraph (c) of this section. Such an agreement shall cover only one escrow accounting year, but a new voluntary agreement may be entered into after the next escrow analysis is performed. The voluntary agreement may not alter how surpluses are to be treated when the next escrow analysis is performed at the end of the escrow accounting year covered by the voluntary agreement. [§1024.17(f)(2)(iii)]

Therefore, if you pay taxes in arrears like I described above and you enter into a voluntary agreement where the borrower will make extra payments in anticipation of higher real estate taxes, at the end of the first year, you'll have a surplus that must be refunded. The borrower can redeposit the funds into the escrow account but you can't make them do so. You can enter into a new voluntary payment shock agreement - covering the 2nd year.

Bottom line: It depends on when the higher taxes will take effect and if your state pays taxes in arrears. You can't just build it into the initial (closing) escrow analysis in all cases.