No relationship. Mixer LO power is a function of the mixer design. If it is totally pasive mixer, the LO power will be a function of desired signal handling capability.
a side, LNA gain of 50 db is very high. If you are able to keep it stable you will likely have other issue with larger nearby frequency signals overloading mixer.
You are better off getting gain after the mixer where you can reduce the bandwidth, thereby eliminating other unwanted signals before amplifing them.