Obviously these Christian jews even obeyed Mose's law until some point, and Paul doesn't say anything to that either but tries to look like he does the same. If I understand this right...
Yes you do. Paul did some odd things himself, even
after he fell off his horse.
Or did the same, and did not tell the Jewish believers not to circumcise their children, etc. There isn't a passage that shows Paul eating a ham and cheese sandwich. Those issues are not addressed.
The early church was nearly all "Jewish Believers". The New Way was for them just like it is for you and I. Ham and cheese sandwiches may well have been a subject of much disputation and indigestion:
Galatians
2:11 But when Peter was come to Antioch, I withstood him to the face, because he was to be blamed.
2:12 For before that certain came from James, he did eat with the Gentiles: but when they were come, he withdrew and separated himself, fearing them which were of the circumcision.
2:13 And the other Jews dissembled likewise with him; insomuch that Barnabas also was carried away with their dissimulation.
2:14 But when I saw that they walked not uprightly according to the truth of the gospel, I said unto Peter before [them] all, If thou, being a Jew, livest after the manner of Gentiles, and not as do the Jews, why compellest thou the Gentiles to live as do the Jews?
2:15 We [who are] Jews by nature, and not sinners of the Gentiles,
2:16 Knowing that a man is not justified by the works of the law, but by the faith of Jesus Christ, even we have believed in Jesus Christ, that we might be justified by the faith of Christ, and not by the works of the law: for by the works of the law shall no flesh be justified.
2:17 But if, while we seek to be justified by Christ, we ourselves also are found sinners, [is] therefore Christ the minister of sin? God forbid.
2:18 For if I build again the things which I destroyed, I make myself a transgressor.
2:19 For I through the law am dead to the law, that I might live unto God.
2:20 I am crucified with Christ: nevertheless I live; yet not I, but Christ liveth in me: and the life which I now live in the flesh I live by the faith of the Son of God, who loved me, and gave himself for me.
2:21 I do not frustrate the grace of God: for if righteousness [come] by the law, then Christ is dead in vain.