Backward prediction

Thing is, this is for a important report and there cannot be any missing data

Philosophical question: If this report is so important, how accurate do these forecasted/extrapolated numbers need to be? In my own field, it sometimes seems better to report data as "missing" or "unknown" than to report some wildly extrapolated number that has not been tested. Only you will know if this is true for your case, but I think it is worth considering.

You don't explain in detail what you have done to "reverse engineer forecasting." If I assume this means you tried using Excel's built in FORECAST() function, you should know that this function is based on a straight line approximation (y=mx+b) to the data. Obviously, not all real world processes follow such a linear trend. However, the process used in the FORECAST() function is a good example of the general process used for this kind of problem:

1) Select a suitable equation. As noted, Excel's FORECAST() function assumes a straight line relationship, so it is only suitable where straight lines are a good approximation of the real world process. There are many other types of functions that can be used (exponential, logarithmic, polynomial, hyperbolic, and so on). This part of the process is not specific to Excel, R, or other application/programming language. I also find that this is often the most difficult part of the process and that computers in general are not good at deciding this step for us. This step will be up to you to make an intelligent choice of equation. FWIW, if I had to guess, it seems that something "total healthcare costs" would follow some kind of exponential relationship, though that is nothing more than a "good guess".
2) Once you have chosen an equation, choose a suitable regression algorithm. Excel has a built in linear least squares algorithm that is used for functions like LINEST(), LOGEST(), FORECAST(), TREND(), GROWTH(), etc. If you decide your equation is suitable for a linear least squares algorithm, then you can use one of these built in functions to perform the regression. If you decide against a linear least squares regression, you will need to program the regression algorithm into Excel yourself. Some of these algorithms can make use of the built in Solver utility to perform the optimization.
3) With parameters obtained by the regression, you can then compute your extrapolated values.

If you have no idea what kind of equation to choose, you could go through a process of trial and error where you select an equation, perform the regression, and see if you like the results. Keep trying different equations until you find one that you like.