As a CPA familiar with the Washington state Business and Occupation Tax (B&O), I can confirm that the tax is both a deductible business expense, and that it appears you have done so correctly by including it in the "Other Common Business Expenses" ->
"Taxes and Licenses" section of TurboTax. That should ensure that the state B&O tax appears on the correct line of your Form 1040 Schedule C (for self-employment).
Thank you for asking this question, and for later figuring out the correct answer for yourself. You've done very well!
