To avoid underpayment penalty you need to pay thru the year via withholding or 'timely' (generally quaterly) estimated tax (ES), the smaller of either - 100% of your 2024 tax (110% if your AGI > 150k), or 90% of your 2025 tax; this is the 'safe harbor' amount. This is inclusive of all your income/withholding not just the Roth conversion. So the first thing to figure out is what's your safe harbor with this additional 12k income event in 2025, whether 2024 method is smaller ES etc.
See https://www.irs.gov/faqs/estimated-tax and https://www.irs.gov/forms-pubs/about-form-2210 for more information on ES etc.
Also confirm if your IRA has any basis from after tax contributions (e.g. non deductible contributions) then your conversion will not be entirely taxable, this would have been recorded on Form 8606 (or if not for whatever reason then you will need to get that filed and taken into account here).
I think your options are then:
1. pay quarterly ES to meet the safe harbor amount, tho you may be late on Q1 payment if you did not already pay that and may result in an underpayment penalty; but the penalty aside you do not have to pay the full ES due at the time of conversion as long as you pay 50% by Q2, 25% in Q3, 25% in Q4. You can earn interest on the Q3/4 amounts you don't need to pay IRS yet which will probably offset the penalty. The annual penalty rate is 8% but this will only be applied on 25% of the ES and only for a few months if you pay it in Q2.
2. pay the full ES in the quarter you do the conversion, however this may require you to file using the Form 2210 "annualized income" method for 2025 to reduce/eliminate any penalty, which will show how the one-off ES lines up with the timing of the income. IRS by default assumes your income from all sources occurs evenly thru the year and hence your ES should ideally be quarterly to align with that, if not this method allows you to specify the uneven timing. The sooner you do the conversion and pay the tax the less the penalty will be that you are trying to eliminate with this filing method. You would have to provide your AGI/withholding etc per quarter (3/31, 5/31, 8/31, 12/31) - and same for state - which can be a lot of additional calculations for your filing depending how complex your situation is.
3. increase withholding if you have W2 income to cover the 'safe harbor' amount, which is always considered 'timely', but you only have 6 months left to cover the extra amount. @VolvoGirl is also suggesting you can do withholding on the conversion itself, but I think that tax then comes out of the IRA money before conversion (your Roth will get <12k) rather than using non-IRA money (your Roth gets 12k) which can be generally disadvantageous to take money out of your tax-deferred/free accounts.
In terms of paying ES once you figure out the amount and approach, you can pay this directly electronically at irs.gov and specify it's for estimated tax and that constitutes your filing you don't need to mail checks/vouchers etc. Your state likely has equivalent electronic payment platform.
Not a CPA just my 2 cents what aspects to consider, hope this helps.
