The way I typically handle the double accent issue while trying to read aloud or speak in a natural-feeling sort of way with a pitch accent is to use relative pitch. So for your example, “Δικαιόπολις Άθηναῖός ἐστιν”, what that means is that the acute accent gets pitched “up”, the syllable following it gets pitched “down”, then the circumflex is an “up” followed by a “down”, the second accent in “Ἀθηναῖός” stays at the same pitch as the end of the circumflex, and then the vowel that follows that acute accent (the ἐ in ἐστιν") is spoken at a lower pitch than the previous vowel.
Now I am no scholar of linguistics so take everything I say here with an appropriate grain of salt. That said, I am a singer and a native English speaker, so I use what I have available to me in my linguistic toolbox to try to make things work.
While English doesn’t have a pitch accent per se, English does regularly use both stress and pitch to convey meaning. Notice, for instance, that in English the italicized word above would be pronounced with stress and the bolded word would be pronounced both with stress and a rise in pitch, with the following word being spoken at a lower pitch. The last syllable of the sentence would be spoken at a lower pitch relative to the preceding syllable in order to indicate that the sentence was declarative. If I used an upward pitch instead, it would indicate an interrogative: “English does regularly use both stress and pitch to convey meaning?” Notice also that if you want to read that sentence out loud in a natural sounding way that your pitch and stress throughout the sentence is entirely different in the interrogative versus declarative mode.
This is very useful, because it gives us a place where you can begin to get a feel for using pitch to convey meaning in a natural way. You already do it, after all, even if you’re not aware of it. And primarily the means by which you do it is by using pitches relative to each other, not ones that are strictly defined in some rigid way.
Pitch is perceived primarily in how it relates to other tones that occur near to it in time. If I sing a C4, and only a C4, it doesn’t really do much. If I sing a C4 and then move up or down a little bit, then all of a sudden that note becomes something meaningful: it can be a fifth, a third, it can be an octave, it can be all kinds of other things. You will perceive the C4 as either “high” or “low” based on what I do either before or after it. I like to use that in the pitch accent by taking what would be really unnatural-feeling constructions like “Ἀθηναῖός ἐστιν” if all the accents were actual “tones” and instead using the mechanics of relative pitch to create the same sort of perceptual experience. In this case, starting from a neutral pitch, you get
-Ἀθ- -η- / ναῖ \ -ός- ἐστιν
Hopefully my formatting makes sense there. Dashes represent neutral pitch, the slashes indicate rising or falling pitch, and the subscript indicates the lower pitch. Essentially what I’m saying is that the second accent in that word can be “ignored”, but should result in the following word being pitched lower than the accented syllable. This causes the listener to perceive the accented “ός” as being pitched “up”, even though the speaker actually hasn’t done so. IMO this successfully conveys the meaning in a much more natural sounding and feeling way that functions quite a lot more like how English “pitch” conveys meaning, which makes it much easier to adapt and use.