Normally, the payer (company) is charged with making that change by submitting a corrected 1099.Any letters that you would prepare is something the IRS won't look at right away. During tax season, they look at forms....so the quickest way to take care of this is to have the company correct the 1099, which is something you don't want to do for fear of how they might handle your information.
I don't think that you actually have the INTENT to fraud or evade, but you have to know that the IRS will be looking for whoever's SS # is on that 1099 to report the income. They can hold that person's refund if they have one, and come after them for understating their income. So again, you're putting someone else through everything you are trying to stop for you
I have attached information from the IRS re: when backup withholding is required, and again, it is required for people who have TIN and NOT for those with SS#. This means that you are not subject to the 28% back up withholding....and since you aren't willing to have the company correct their records so that your income is under the correct SS#, I'm confused as to why you would want the company to pay in the 28%. This 28% gets added on to your 1099 as part of your Misc income on the 1099 you will receive, so if you had $10,000 in Misc income, and the company sent in the 28%, your 1099 would show $12800 in income with $2800 as your backup withholding.
Again, I'm confused by your insistence at having the employer send this withholding in for you. It has missed the deadline for tax year 2016, so it would only count for 2017, and you stated that you no longer work there.
I'm happy to help you, but I'm not entirely sure where you want to go with this.
Here's the attachment I mentioned: