Hmm.... So I know there are a few household cleaners that are safe, but off the top of my head I can't think of them BUT most you still have to use water to rinse off. Are you wanting something to clean the cage and keep the bird in it while you do so?
Although citrus fruits are not recommended for birds to eat regularly, small amounts are not poisonous, and many seed diets add orange oil for flavor, scent and as an acidic preservative. BUT besides that, you can used lemons, oranges and limes to clean the cage. It's acidic enough that it breaks down goopies, and leaves a fresh scent AND if your bird chews on metal or plastic that has the fruit juice dried on it, it's safe.
I'd say slice up lemons into wedges, pull out the seeds and scrub away; you'll make a drippy mess, but it's an edible drippy mess vs any chemical sprays that would drip too.
You can try it out n let us know how it goes.....
There are also bird poop cleaners safe for birds, that are not household items if all else fails. Ex.Poop Off and the Bird Cage Cleaner made by Nature's Miracle