What then shall we say was gained by Abraham, our forefather according to the flesh [κατὰ σάρκα]? 2 For if Abraham was justified by works, he has something to boast about, but not before God. (Rom. 4:1–2, ESV)
While the proximity of κατὰ σάρκα in Romans 4:1 to Ἀβραὰμ τὸν προπάτορα ἡμῶν makes τὸν προπάτορα ἡμῶν seem to be the modified, this has a problem in the context, because in this case κατὰ σάρκα eliminates "we" including the Gentiles. Thus, this seems to be inconsistent with the context.
The previous context:
For we hold that one is justified by faith apart from works of the law. 29 Or is God the God of Jews only? Is he not the God of Gentiles also? Yes, of Gentiles also, 30 since God is one—who will justify the circumcised by faith and the uncircumcised through faith. 31 Do we then overthrow the law by this faith? By no means! On the contrary, we uphold the law. (Rom. 3:28–31, ESV)
The context that follows:
9 Is this blessing then only for the circumcised, or also for the uncircumcised? For we say that faith was counted to Abraham as righteousness. 10 How then was it counted to him? Was it before or after he had been circumcised? It was not after, but before he was circumcised. 11 He received the sign of circumcision as a seal of the righteousness that he had by faith while he was still uncircumcised. The purpose was to make him the father of all who believe without being circumcised, so that righteousness would be counted to them as well, 12 and to make him the father of the circumcised who are not merely circumcised but who also walk in the footsteps of the faith that our father Abraham had before he was circumcised. (Rom. 4:9–12, ESV)
Supporting εὑρηκέναι being modified: in Galatians Paul used κατὰ σάρκα to refer to human effort, which in that case resulted in Ismael.
But the son of the slave was born according to the flesh [κατὰ σάρκα], while the son of the free woman was born through promise. (Ga 4:23, ESV)
Note Paul's reference in Rom. 4:13 to promise:
For the promise to Abraham and his offspring that he would be heir of the world did not come through the law but through the righteousness of faith. (ESV)
Note in Galatians:
And if you are Christ’s, then you are Abraham’s offspring, heirs according to promise. (Gal. 3:29, ESV)
not κατὰ σάρκα