Psalm 51:4 (v.6 in Hebrew)
In BHS1 v.4 is v.6, and the construction in Hebrew is an infinitive construct with both a prepositional prefix and a 2nd person singular pronominal suffix attached (בְשָׁפְטֶֽךָ).
The pronominal suffix can be used on an infinitive construct as either a subject or object of the verb.2 That means there is flexibility in the construction itself that allows for it to essentially mean "God [doing] judgement" or "judgement [done to] God." This is true of the previous construct as well, which could mean "God speaks" or "speaking [about] God."
The common practice of Hebrew parallelism is probably partly what drives the typical translation of God as subject in both cases (that keeps it parallel). Additionally, theologically, one is left to ask how it is that God is judged and by who (which is the essence of your question here).
The context of Psalm 51:4 shows a "judging" of God occurring. David is "judging" that God is right, David has sinned and done evil (vv.3-4).3 Basically, in every action a person takes, four judgments are occurring—(1) God is judging the righteousness of a person's actions, and (2) a person is judging the rightness of his own actions. Thus in the outworking of (2), then (3) a person is judging God's words about what He has revealed to be right (if a person is agreeing, the person considers God "right," if disagreeing, the person considers God "wrong"), and (4) God is judging the person's judgment of His own character, whether they acknowledge Him as "righteous" or not by being in agreement with Him.
This is actually still expressed in a parallel form in the verse if the second statement is considered object, not subject. Note the chiastic thought structure (A B B' A') of the verse then the ESV text given best leaves ambiguity for the last part of the verse being subjective or objective, but here it is assumed objective:
Against you, you only, have I sinned [David making right judgment of God's words]
and done what is evil in your sight, [David referring to God's judgment of him]
so that you may be justified in your words [David referring to God's judgment in general]
and blameless in your judgment. [David referring to anytime God is judged]
This idea of God's words being judged is carried over to Romans 3.
What is interesting is that God through Paul4 does not hold it the concepts as parallel in Romans 3:4. Indeed, Paul converts the first infinitive ("You speak") in the Hebrew into a noun in the reference (τοῖς λόγοις σου; "the words of you"), but keeps an infinitive for the second, but specifically makes it middle/passive in form (it could have just as easily been made active if the idea was intended to be God judging, rather than God being judged).
The best support for it not being a middle voice is the context of Romans 3 itself. The lead up to v.4 is v.3, where the question is referring to "the faithfulness of God" (NKJV) in light of people's (specifically in context, Jew's) unbelief. All unbelief is simply a manifestation of that person's judging God's revealed and recorded words (the subject of v.2) to be false.
However, v.4 is asserting that any person's judgment of God as false, resulting in unbelief, is no proof at all that God is not true and righteous. He is true, His words true, and He is righteous, no matter if every man lies about God otherwise and does not believe His words.
So "who is doing the judging" of God (and specifically God's words)? All people. Those judging Him true judge rightly, and believe Him. Those judging Him false judge wrongly, and do not believe Him.
"Does Paul elsewhere portray God as the object of judgement?" Yes and no. Yes, when one realizes that belief/unbelief is such a judgment, then the Bible (not just Paul) is full of instances of God's being judged. No, I could not find a use of κρίνω with God as object elsewhere in Paul's writings.
However, Rev 16:7 makes a statement of judgement of God (likely by an angel):
“Even so, Lord God Almighty, true and righteous are Your judgments.”
1 Biblia Hebraica Stuttgartensia: With Westminster Hebrew Morphology Electronic ed. (Stuttgart; Glenside PA: German Bible Society; Westminster Seminary, 1996), Ps 51:6.
2 Ronald J. Williams, Williams' Hebrew Syntax, 3rd. ed. (Toronto: University of Toronto Press: 2007), 48.
3 The context is David's repentance after hearing God's judgment against David by the prophet Nathan for his sin with Bathsheba.
4 I hold to a hermeneutic that all Scripture is ultimately of divine Authorship, so both Romans (through Paul) and Psalm 51 (through David) have the same ultimate Author behind them.