I don't think Putin gives a damn about public pressure. The only public opinion that matters to Putin is the one that he, himself, so generously gives to the people to help enable the State to accomplish its great and glorious mission. Ergo, the real question is how does Putin see the presence of the SAS in western Ukraine?
If you come at the problem from the perspective that every right-thinking Ukrainian must want Russian hegemony, then clearly anyone who fights against such rule must be (a) a nazi, or (b) a tool of NATO which is using those tools to destroy Russia (other bêtes-noires are available). I expect Putin to play the presence of the SAS as an example of the latter, and "inflaming" the conflict to prevent Russia from achieving its "righteous" goals.