I've been researching this for some time now and I got stuck.
The best resource I've found is this book. The "optimising a multistage rocket" part talks about a three stage rocket but it seems applicable here. The part where I got stuck is at the page 29, where author represents fractional mass of the three stages. Could someome please explain the steps after that to me ? His "simple manipulations" don't seem as simple to me :D
Other solution I wanted to try was to modify the Tsiolkovsky rocket equation for my two-stage rocket into a function with a variable being the ratio between the stage 1 and 2. This function's maximum would be the desired ratio. But my high school mathematics obviously weren't enough for that, because I got nowhere.
Could you please nudge me in the right direction ? Thank you very much.