Hi Zapa_92
In a normally functioning Mac with a missing or broken mouse you can navigate around the various nodes on the active window by using the Tab key, the arrow keys, the space bar and the enter key.
This Apple Doc explains in greater detail:
https://support.apple.com/en-gb/HT204434
The problem is that you are interfacing with the Mac at a very low level and I don’t know if these special keyboard shortcuts and commands are all functional in this recovery mode.
If the above is of no help then try and borrow any standard USB mouse from a Windows or Mac and they will work, your Logitech is a bit different because they use customisable drivers with non-standard functions, which is why it won’t work for you now, but virtually any other working mouse should get you by this road block.
Will.