It doesn't appear that CiviCRM can send an outbound SMS via an activity on a case. The outbound SMS activity shows as a normal activity, rather than the send SMS screen. Is this normal / expected behaviour and are there work arounds to support SMS on a case?
It's probably just never come up. There is special handling in the code for Email to make that work. It would need something similar.
However, by somewhat of a fluke, you can use the subject line feature for activities to send a regular SMS the normal way and have it filed on a case. Where it says "Name the SMS", include
[case #1234] somewhere in that field, where 1234 is the appropriate case id. Then it will file it on the case.