I'm going to post two more early 20th century or earlier African American rhymes that mention policemen. These two are from Thomas W. Talley's 1922 book Negro Folk Rhymes, Wise and Otherwise. I have the 1968 Kennikat Press edition of that book, and the page numbers cited will be from that edition. I'll type the one rhyme in this post, and the next rhyme in the following post to this thread.
I may have posted these rhymes on other Mudcat threads, but I can't find them.

The Negro And The Policemen

"Oh Mistah Policeman, tu'n me loose;
Hain't got no money but a good excuse."
Oh hello, Sarah Jane!

Dat ole Policeman treat me mean,
He make me wa'k to Bowlin' Green.
Oh hello, Sarah Jane!

De way he treat me wus a shame.
He make me wear dat Ball an' Chain.
Oh hello, Sarah Jane!

I runs to de river, I can't git 'cross;
Dat Police grab me an' swim lak a hoss.
Oh hello, Sarah Jane!

I goes up town to git me a gun,
Dat ole Police sho' make me run.
Oh hello, Sarah Jane!

I goes crosstown sorter walkin' wid a hump
An' dat ole Police sho' make me jump.
Oh hello, Sarah Jane!

Sarah Jane, is dat yo' name?
Us boys, we calls you Sarah Jane.
Well, hello, Sarah Jane!

[page 66]