Top contenders:

Red-cockaded Woodpecker — “cockaded” refers to a ribbon or rosette ornament once worn on hats, not anatomy.

Cock-of-the-rock — sounds like a tavern name invented by a teenager.

Dickcissel — often cited as the funniest North American bird name. “Dick” was historically a common nickname for a male bird.

Bush Thick-knee — not penis-related, but frequently gets laughs.

Rufous-naped Lark — harmless, but “rufous-naped” is often misread at a glance. Shag — in British English, perfectly normal; elsewhere, not so much.

Cockatoo — contains “cock,” though the name comes from Malay, not English.

Woodcock — another classic.

Black-cockatoo and other cockatoos — bonus points for stacking “cock” into longer names.