How do Roman Catholics understand salvation in light of Romans 3&4? As far as I can understand it, Paul directly contradicts the Roman Catholic doctrine of salvation in Romans 3 &amp; 4 (see verses 3:21-22, 3:28, 4:4-6 in particular, but I'm referring to the broad argument of the whole section).

How do Roman Catholics interpret what Paul is saying here?

For clarification, I understand that Catholic doctrine is nothing if not nuanced, and it's not accurate to just dismiss it as "works salvation". Salvation on that view is "by grace alone", but (as far as I understand) not "by faith alone" - rather, God gives us grace to perform the meritorious works that earn our salvation. It's the "faith alone" part that I think Paul is hammering home here.
Essentially it boils down to that we are saved by grace alone, but that salvation results in such a change of heart that good works must result.
From the Joint Declaration on the Doctrine of Salvation (all emphasis mine):

27.The Catholic understanding also sees faith as fundamental in justification. For without faith, no justification can take place. Persons are justified through baptism as hearers of the word and believers in it. The justification of sinners is forgiveness of sins and being made righteous by justifying grace, which makes us children of God. In justification the righteous receive from Christ faith, hope, and love and are thereby taken into communion with him. This new personal relation to God is grounded totally on God's graciousness and remains constantly dependent on the salvific and creative working of this gracious God, who remains true to himself, so that one can rely upon him. Thus justifying grace never becomes a human possession to which one could appeal over against God. While Catholic teaching emphasizes the renewal of life by justifying grace, this renewal in faith, hope, and love is always dependent on God's unfathomable grace and contributes nothing to justification about which one could boast before God (Rom 3:27).

and later in the same document:

37.We confess together that good works - a Christian life lived in faith, hope and love - follow justification and are its fruits. When the justified live in Christ and act in the grace they receive, they bring forth, in biblical terms, good fruit. Since Christians struggle against sin their entire lives, this consequence of justification is also for them an obligation they must fulfill. Thus both Jesus and the apostolic Scriptures admonish Christians to bring forth the works of love.
38.According to Catholic understanding, good works, made possible by grace and the working of the Holy Spirit, contribute to growth in grace, so that the righteousness that comes from God is preserved and communion with Christ is deepened. When Catholics affirm the &quot;meritorious&quot; character of good works, they wish to say that, according to the biblical witness, a reward in heaven is promised to these works. Their intention is to emphasize the responsibility of persons for their actions, not to contest the character of those works as gifts, or far less to deny that justification always remains the unmerited gift of grace.