Er, I don't think I understand. The 26" measurement refers to the diameter of the rim, the 1.25 and 1.5 measurements refer to the depth of the tyre. There should be no problem fitting a 26x1.25" tyre to a 26" rim that previously took a 26x1.5" tyre.
It isn't unusual for me to have some difficulty installing tyres with my bare hands, and I have reasonably strong hands. I don't think you should conclude that you have the wrong size of tyre.