Present competing offers from other time-tracking software as leverage in negotiations with Time Doctor. Indicate to Time Doctor that another vendor has quoted a lower rate for similar functionality, which your finance team is insisting on to proceed with the purchase.
If you plan to reduce the number of users or desired functionalities in your contract with Time Doctor, leverage that descope by communicating to them that you expected rates to decrease, not increase, as your usage is likely to change, causing your budget to be tighter this period.
If Time Doctor presents an uplift in pricing, address that you expected a flat renewal based on last year's expenses. Justify the need to avoid any increase by focusing on current usage and your financial limit, indicating that your organization cannot accommodate uplift without clear ROI.
If your organization is planning to increase the number of users leveraging Time Doctor, leverage this by stating that your growing demand should be met with proportional discounts for unit pricing. This will help initiate discussions for lower per user costs in your negotiations.
If Time Doctor proposes a standard annual rate without acknowledging any previous discounts, push back by explaining finance's budgeting allows only for renewing at the previous year's rates. Argue that past rates were presumed to carry through rather than assume a one-time discount had been applied.