When you open the Quick Use box on google/com/fonts, there is a column on the left that says “Choose the styles you want”. For Lato, the list includes 10 different styles, but only Normal 400 is checked. Check all that you want to be loaded. Then add the font to via the Weaver Xtreme font control option. (Delete the old Lato you had first.)
BUT – when you use specific “sets”, you will have to provide your own styling – either with a custom style, or an inline style. For example, to use Lato thin italic, you’ll have to define a class such as:
.lato-thin-italic {font-weight:100;font-style:italic}