Your assertion that Paul is redefining the meaning of Israel is false. Paul is making clear the Abrahamic covenant can't be transfered to someone else for fulfillment. He points out that once a covenant has been ratified it can't be changed in any way. This includes provisions and the parties involved. The covenant was made with Abraham and his descendants, and it remains that way.
In spite of your claim that the Church is the "true Israel" the facts of Scripture don't support you. The writers of the New Testament consistently make a distinction between the two. The term Israel is used over seventy times in the New Testament, and in each case referencing ethinic Israel.
Let's look closer at the Scripture text you've quoted as proof of your assertion.
"For neither is circumcsion anything, nor uncircumcsion, but a new creation. And those who will walk by this rule, peace and mercy be upon them, and upon the Israel of God" (Gal. 6:15-16).
Your conclusion couldn't be based on anything other than a mistranslation of the word "and" (the word that appears before the term "Israel of God"). In order for your theology to be Scriptural you have to set aside the primary meaning of "and" in favor of the secondary meaning of "even."
Everyone is in agreement that "them" refers to believing Gentiles and using "even" makes the passage read, "mercy be upon them, even the Israel of God." This translation essentially equates believing Gentiles with the Israel of God, but the intepretation is grammatically weak and fails to take into consideration the other uses of Israel in the New Testament. Also, it doesn't deal with the entire context of Paul's discussion with the Galatians. (Verse 16) concludes a letter that warns against Judaizers. The Church was made up of both Jews and Gentiles, but Judaizers were attempting to persuade the Gentiles of the need for circumcsion in order to be saved. Needless to say Paul was in opposition to this teaching, pointing out that everyone was saved the same way, either Jew (the circumcsion) or Gentile (called the uncircumcsion). When he comes to the end of his letter he pronounces a blessing upon those same two groups. Therefore, the word "them" refers to believing Gentiles and the "Israel of God" refers to the believing Jews who were in the Galatian Church.
Examination of (Gal. 6:16) and the rest of the New Testament shows there's no evidence that "Israel" is ever used for anything other than the entire nation or the believing Jews within the nation.
In Christ,
Tracey