"Under the Law": A Look at Paul's Ambiguous Referential Language

Read this today and would like to know what you all think about it:

"[Paul uses] one group identifier ['the Jews'] in [1 Corinthians 9:20a], and another Jewish group identifier ['those under the law'], in [1 Corinthians 9:20b].  In this social categorization, 'the Jews'...describes the vast majority of Jews living around the Mediterranean basin, while 'those under the law'...represents a subgroup identity within this broader classification of Jews....
Paul uses the phrase ['under the law'] four times in 9:20b, which raises the following questions:  To whom does Paul refer by the group descriptor, 'those under the law'?  How does Paul become 'as one under the law'?  What does he mean when he claims that he is 'not under the law'?  Finally, what does it mean to 'win those under the law'?
Schrage claims that ['under the law'] refers to 'Jews living under the authority of Mosaic law.'  However, the use of this phrase to simply refer to Jews is redundant and would not explain why Paul would use it right after writing ['the Jews'] in the first part of verse 20.  This suggests that Paul has a different referent in view here.  Ciampa and Rosner contend that it refers more specifically to 'proselytes.'  While this is possible, the phrase ['under the law'] is quite ambiguous in Paul's letters.  For example, in Gal 3:10, 13 ['under the law'] can mean 'under the curse of the law'], while in Rom 6:14-15, it can be understood as 'under the sin-strengthening regime of the old age.'...
Lightfoot and Bockmuehl have suggested another reading for ['under the law'], one that refers either to those holding to a more 'strict interpretation of the law' or more specifically to the 'Pharisees.'  This view is preferred for the following reasons:  First, in Phil 3:5, Paul says of himself ['as to the law, a Pharisee'.  This makes for a good conceptual parallel to the language of 1 Cor 9:20.  Second, Pharisees were known for a more strict observance of the law compared to the halakhah of Jesus.  Third, this understanding distinguishes the small group from the previously mentioned larger group, i.e., 'the Jews.'  This distinction between the Pharisees and the rest of the Jews is furthermore evident in Mark 7:3 ('the Pharisees and all the Jews') and Luke 7:29-30, as well as Josephus JW 1.110.  So this proposed understanding of ['under the law'] as 'those following a [strict interpretation] of the law' would suggest that Paul no longer follows the Pharisee sect....
So, in answer to the questions raised by the presence of ['under the law'] in 9:20, Paul uses the group descriptor, 'those under the law,' to designate those living under a strict halakhic interpretation of the Pharisees.  How does Paul become 'as one under the law'?  He does this by sharing table-fellowship with those who hold to these strict standards, and he, himself, follows them on those occasions.  What does he mean when he claims that he is 'not under the law'?  Paul is no longer following the strict halakhot of the Pharisees.  Finally, what does it mean to 'win those under the law'?  It could mean to convince those in this group that gentiles in Christ are not required to follow similar halakhic standards; rather, they are to keep the commands appropriate to them (1 Cor 7:19)," (pg. 102 of Remain in Your Calling:  Paul and the Continuation of Social Identities in 1 Corinthians by J. Brian Tucker)