I tried to experiment with BINOM.INV(), but I never got anything even close to the right solution with this function.
As I said, that is because BINOM.INV() is inverting the cumulative distribution function, not the probability density function. If you know what your target values for the PDF translate to on the CDF, then you can easily use BINOM.INV(). I don't know how to translate the PDF to CDF in a case like this, so I don't know how doable that approach would be.

My solution to the "I could use Solver but I don't want to use Solver" scenario is to program my own "root-finding" algorithm. I have a tutorial here: https://www.excelforum.com/tips-and-...ind-roots.html The main first question is whether to do it purely in the spreadsheet (with iteration enabled) as shown in the first several examples. The final example (post #9) shows how this might be done in a VBA user-defined function.

How do you want to proceed?