You are not required to submit the worksheets. There is no tax or business reason why the employer needs to know how you decided how many allowances to claim, and it is an invasion of your privacy for them to demand to see such information as your spouse's salary and your expected medical expenses or other deductions.
https://www.irs.gov/compliance/criminal-investigation/employer-and-employee-responsibilities-employm...
https://www.irs.gov/taxtopics/tc753
If you claim exempt, or claim more than 10 allowances, the employer may be required to send the form to the IRS, and the employer may have some responsibility to see that your claim to be exempt from withholding has a factual basis. But normally, they aren't required to keep the worksheets or have them available to the IRS.
Unfortunately, as a practical matter, this is going to come down to who is more stubborn, and whether your employer needs you more than you need them. If you simply refuse, will they fire you? If you refuse to provide a W-4 they are required to withhold as single with no exemptions, maybe you do that and adjust your spouse's withholding accordingly so you don't have too much withheld overall. Or maybe you tell them you will submit the worksheets if their CPA writes a letter giving the section of the IRS regulations that requires the employer to keep a copy (which does not exist.) I doubt the IRS will do anything if you call them, but your state labor department may be more willing to act on a phone call and tell the employer to stop violating their employees' privacy.
