Since you say that you need to withdraw more than your contribution basis and the amount that you withdraw in excess of your contribution basis will be subject to a 10% early-distribution penalty, I'll infer that the need to obtain a distribution has nothing to do with an excess contribution.
The amount that you want to have withheld for taxes depends on your objective. Objectives would be to avoid underpayment penalties, to avoid having to pay more with estimated tax payments or a balance due on your tax return, or to avoid needing to annualize income on Schedule AI of Form 2210 to eliminate an underpayment penalty for earlier tax quarters.
Only the amount that you withdraw in excess of your $1,200 of contribution basis is subject to tax and, assuming no Roth conversion basis, only taxable amounts are subject to the 10% penalty. If you are trying to have the tax withholding exactly match your increase in tax liability, you would need to determine your federal marginal income tax rate based on everything else on your federal tax return and add 10% for the penalty. Utah has a flat tax rate of 4.85%, so your marginal state tax rate would be no more than that. Ideally you would create a simulated tax return to determine your incremental tax that results from the distribution, which is most easily done with the CD/download version of TurboTax.
Given that it's so close to the end of the year, you'll need to decide if you want this distribution to be 2023 income (with tomorrow, December 29, 2023 being the last day to be able to obtain such a distribution) or 2024 income (with the distribution occurring next week or later).
Did you make a contribution for 2023? If so, it might be beneficial to obtain a return of contribution instead of a regular distribution to avoid some or all of the early-distribution penalty.
Given that your Roth IRA contribution basis is $1,200, it seems a bit unusual (although not impossible, particularly if you have basis in taxable Roth conversions) that you would have much more than $1,200 available to distribute, so does it even matter much how much you have withheld for taxes?
