You are switching approaches, a LOC Approach is seperate from an ILS Approach. They have different minimums, different MAPs, different FAFs, one's precision and the other is not. They share the same frequency, that is true, but that does not make them the same approach.

Whether or not you should switch from one approach to the other if the GS fails, well that is up to you and the current situation you are in, and whether you are capable of switching to the other approach, or whether it would be safer to go missed and setup for another approach.