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"billing," OED, " The caressing of doves; kissing; love-making." Gheti Afroz is using this adjective to reproach Melech Mahommed for the forwardness of his advances. This was an adjective often associated with doves in the 18th and 19th centuries--well within Leyden's time period-- though it seems to have fallen out of use after his time. These connotations of unrestrained demonstrative affection would have furthered the connection between Melech Mahommed's "crime" (i.e. behaving like a billing dove) and "punishment" (being changed into one.)
[[http://www.oed.com/view/Entry/19020?rskey=Z1IKeX&result=2&isAdvanced=false#eid.=OED, 2. "billing," ]] 
However, it is still a relatively "honorable" animal to be associated with, since Islamic tradition esteemed the dove for once helping the prophet Muhammad.