Present participle of cooee; making cooee calls, or continuing to attract attention with the cooee sound.
From cooee plus the -ing suffix, following standard English progressive verb formation rules.
The spelling cooeeing with three e's in a row shows the collision between Aboriginal phonetics and English spelling rules—it looks awkward to English eyes but it's the logical result of adding -ing to a word that ends in -ee.
Complete word intelligence in one call. Free tier — 50 lookups/day.