First, you don’t have to be in pose mode for weight painting to work. Just click armature and then shift+click mesh.
Second, if you want to do pose mode option then you can shift+alt+left click on mesh to get a list of objects under the mouse cursor. Assuming you’re not using right click select or emulate 3 button mouse.
Third, if the list doesn’t come up or you are using emulate 3 button mouse or you just don’t want to do the list then the issue could be GPU depth selection which can be turned off by unchecking it in preferences->Viewport->selection
