My clothesline is under a big tree full of poopy birds too, so I feel your pain. LOL!
Our clothesline is one of those old-fashioned T-post kinds with five wire lines. So I don't know if this would work for you, but my (cheap!) solution was to clothespin two old twin flat sheets over the whole thing like a canopy. I pinned them to the two outer lines and use the three inner lines for the laundry. I hope that makes sense how I described it.
I've been really pleased with it. No more poop on my laundry! And the side benefit is that it keeps the dark laundry from getting sun faded. I haven't bothered to wash the sheets turned canopy yet, but I suppose I'll do it every once in a while. The nice thing is the T-posts are quite tall so I can stand up fully under the canopy.
Could you possibly hang three lines in your yard, then hang an old sheet on the outer lines and use the inner line for your laundry? Just hang your lines pretty high so you can stand under the canopy.