void lsm_dom_view_set_cairo (LsmDomView *view, cairo_t *cr)

Is a misnomer - there is no such thing as a 'cairo' - cairo_t is a cairo context. See http://cairographics.org/manual/language-bindings.html

Suggest deprecating the old name and adding lsm_dom_view_set_cairo_context()

Or as a slightly bigger, but better, change, pass the cairo_t as a parameter to lsm_dom_view_render - the cairo context object is a transient object holding the current state of a rendering process. Setting it as an object property never really makes sense.