Cross-pollinating cannot change the character of any fruit. It only affects the genetics of the seed being produced.

There are 3 flower types of papaya. The flowers are either male, female or both in one flower (self-pollinating).

I grow 'Red Lady' because it is a dwarf and self-pollinating. I believe that 'Solo' needs a pollinator. So, that means that you need a male plant nearby, (doesn't matter what the male variety is). I am not familiar with the Mexican Red.