When we consider the whole of Paul's teaching about the law of Moses there's no reason to think he meant faith upholds the letter of the law because he's quite clear about various first covenant stipulations and governance being laid aside because the way of faith has appeared. And we know the priesthood was allowed to pass rabbinical decisions down, adding them to the law. The allowance for divorce, for example, was actually a rabbinical add-on to the law of Moses. So, in the end, we know the establishing of the law does not mean the establishing of the letter of the law. And we know that the Spirit filled life actually takes the standard of righteousness above and beyond the righteousness required in the law.
But the reason I think Paul is saying faith upholds the law of Moses is because that is the very context of the passage where he says that:
"19 Now we know that whatever the Law says, it speaks to those who are under the Law, so that every mouth may be closed and all the world may become accountable to God; 20 because by the works of the Law no flesh will be justified in His sight; for through the Law comes the knowledge of sin.
21 But now apart from the Law the righteousness of God has been manifested, being witnessed by the Law and the Prophets, 22 even the righteousness of God through faith in Jesus Christ for all those who believe; for there is no distinction; 23 for all have sinned and fall short of the glory of God, 24 being justified as a gift by His grace through the redemption which is in Christ Jesus; 25 whom God displayed publicly as a propitiation in His blood through faith. This was to demonstrate His righteousness, because in the forbearance of God He passed over the sins previously committed; 26 for the demonstration, I say, of His righteousness at the present time, so that He would be just and the justifier of the one who has faith in Jesus. 27 Where then is boasting? It is excluded. By what kind of law? Of works? No, but by a law of faith. 28 For we maintain that a man 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 indeed God who will justify the circumcised by faith and the uncircumcised through faith is one. 31 Do we then nullify the Law through faith? May it never be! On the contrary, we establish the Law." (Romans 3:19-31 NASB)
For him to be referring to the law of Moses specifically in all these places where he says 'law', but to then suddenly mean a different 'law' in the last sentence is grammatically and logically unreasonable. The bottom line is, we know Paul is not talking about faith upholding the letter of the law, but rather the righteousness of the law.
The following illustrates how unreasonable it is to say Paul means a different law of God other than the law of Moses is what gets established by faith. Surely you agree 'the law' in the two sentences in verse 31 have to be the same law, so let's see how awkward it becomes if we change the law that gets 'established' to a different law than what gets 'not nullified':
"31 Do we then nullify the Law (of Moses--the law that can not justify) through faith? May it never be! On the contrary, we establish (a different law) the Law (of God)."
See? The law that may never be nullified by faith has to also be the law that gets established by faith. Otherwise the verse makes no grammatical or logical sense. We just need to remember he's talking about the righteousness of the law, not the letter of the law.