Everything an employer pays an employee must be reported on their W-2, and is subject to income tax withholding and social security and medicare tax. This was never a tax-free gift, and the new owner is avoiding his share of social security and medicare (accidentally or on purpose) by reporting it on a 1099 instead of the W-2.
Start by asking the employer to cancel the 1099-NEC and issue a corrected W-2. If they refuse (keep copies of your correspondence), then your husband should file by entering the 1099-NEC and checking the box for "I received a 1099 and a W-2 from the same employer and this income should have been on my W-2." Turbotax will prepare a form 8919 with code H, to collect the employer half of social security and medicare (7.65%) instead of self-employment tax (15.3%). If a bunch of people do this, the employer will probably get a call from the IRS.
I do think you are kind of overreacting. While it may be a surprise that you owe $500 or $600 in taxes, you will still net $1400 or so from the bonus. And TurboTax can prepare form 8919 for you, there is a question that covers this exact situation. If this requires an upgrade to a paid version of TurboTax from the free version, and you don’t want to pay the fee to use the paid version, you may be able to find other tax software that can prepare the appropriate form. Or you can prepare your tax returns by hand. You can download all the forms and instructions, including form 8919, from the IRS website.
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