Exegetically speaking, the subjective genitive translation "faith(fulness) of Jesus Christ" in Romans 3:22 has problems. It is often said that Romans 3:3 refers to the "faithfulness of God," and therefore, Romans 3:22 is referring to the "faith(fulness) of Jesus Christ," but this conclusion ignores a couple big problems. First, it is not clear that Romans 3:3 should be translated "faithfulness of God." There seems to be a sort of play on words happening in Romans 3:2-3 where the same root word in the Greek is used four times in close proximity. The Israelites were (1) "entrusted" with the oracles of God, but some of them did not respond with (2) "trust." However, their lack of (3) "trust" did not nullify the (4) "trustworthiness of God." Therefore, "let God be true and every man a liar." If this is the correct interpretation of these verses, then the argument for the "faithfulness of God" in Romans 3:3 falls to the ground and cannot be used as evidence for translating Romans 3:22 as "faith(fulness) of Jesus Christ." However, even if "faithfulness of God" is the correct interpretation for Romans 3:3, it is far removed from Romans 3:22 and appears in a completely different context. The "faithfulness of God" in Romans 3:3 and the "righteousness of God" in Romans 3:5 are in the context of God taking vengeance and judging the world. This context is very different from the salvific activity suggested by those who want to translate Romans 3:22 as "faith(fulness) of Jesus Christ" and interpret "righteousness of God" as "God's covenant faithfulness."
Moving on to Romans 3:22 specifically, the immediate context presents the belief of the believer, and this is consistent with the belief and faith of the believer in the thematic verses of Romans 1:16-17. This context and backdrop support the objective genitive translation "faith in Christ."
Also, if Paul does intend to teach the "faith(fulness) of Christ" in Romans 3:22, then it is surprising that he never unpacks this ambiguous statement. In the context, we learn that we are justified by the death of Christ in His blood (Romans 3:25 and later in Romans 5:6-10), but if this is all that the "faith(fulness) of Christ" means, then it doesn't permit any speculation about "Christ's faithfulness" working in the believer. The word pistis (i.e., faith) does not even appear once in Romans 6 to 8, so Paul's discussion on sanctification in those chapters is certainly not meant to unpack the meaning of the phrase "faith(fulness) of Christ." And yet, many of the proponents of the "faith(fulness) of Jesus Christ" translation (such as Douglas Campbell) insist that those chapters comprise the heart and soul of the gospel.
Commenting on this issue, Moises Silva said the following in Volume 3 of the New International Dictionary of New Testament Theology and Exegesis (see especially page 769):
With regard to the debate as a whole, I happen to believe, naively perhaps, that the evidence is not all that ambiguous—or to put it more accurately, that the ambiguities in the data are plainly resolved by Paul's many unambiguous statements. If by pistis Christou (which in isolation can indeed signify any number of things) the apostle had meant either "Christ's faith" or "Christ's faithfulness," it would have been ridiculously easy for him to make that point clear beyond dispute. Among various possibilities, he could have, for example, indicated—in the same contexts—one or two ways in which Jesus believed and how those acts of faith were relevant to the matter at hand. Or he could have told us—again, in the same contexts—that his message of dikaiosyne ("righteousness, justification") is true because Christos pistos estin ("Christ is faithful"). What could have been simpler? And considering the theological importance of this issue, one would think that he might have made a special effort to clarify matters.
Instead, if some scholars are to be believed, Paul did not have enough sense to realize that the phrase pistis Christou is ambiguous. And to make matters worse, he unwittingly misled his readers by using the verb pisteuo with Christos as direct object again and again in the very same passages that have the ambiguous phrase! His bungling proved spectacularly successful, for in the course of nearly two millennia, virtually every reader—including ancient scholars for whom Greek was their native language—understood the phrase to mean "faith in Christ" and gave no hint that it might mean something else.