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Kentucky Department of Revenue here states:

The Department of Revenue does not administer local occupational, net profits or gross receipts taxes. Should you have any questions concerning any local tax, please contact the appropriate local official indicated at the Kentucky Occupational License Association.​

The Commonwealth of Kentucky permits local taxing authorities to tax income that is sourced within their physical jurisdiction.  Secretary of State website is here.

In the aftermath of the 2020 COVID experience, I am aware of employees who received a refund of local occupational taxes paid and/or were allowed to be exempt from the local occupational taxes when they live and work remotely.

Contact your employer's Human Resources office to see whether their office will assist you in removing this tax assessment from your paycheck.

You are probably going to have to contact the tax authorities at City A for a definitive answer.  Even if they are allowed to tax you, City B probably gives you a credit for what City A withheld. 

I'm in Ohio and the rule here is that City A can't tax you in that situation.