This is a difficult question to answer briefly in this forum as most, if not all, of the class level metrics are designed to address different aspects of fragmentation. I don't want to recommend specific metrics since that really depends a lot on the specific objectives and the user and ease of interpretation. However, I would suggest, given your broad interest fragmentation in general, to pick one or two metrics from each group of metrics and then, post-hoc, look for both consistencies and redundancies. In the end, I have found that a minimum of, say, 6 metrics and a maximum of, say, 12 metrics would be useful in the end for such an analysis. Also, I wouldn't dismiss the edge metrics outright. Remember, many of the metrics either directly or indirectly deal with edge effects. You are correct that habitat is a species-specific phenomenon, but that would apply to all habitat fragmentation metrics. Clearly, edge depths for the core area metrics would be difficult to apply without a clear species or ecological process in mind, so that might be the only group that I might dismiss, but even that group could be used with a fixed edge depth to assess the fragmentation from a comparative standpoint. Anyways, just a few thoughts to help you get started, but ultimately there is no short cut around getting familiar with the various metrics before conducting the final analysis.
