First, I suspect you are SoL. Second, any change will be a long time coming, and you are one foot in the hole in that you are already in violation, seeking a legislative change rather than seeking a zoning exemption before beginning this endeavor.
I've helped write legislation in many States, opposed some, helped redraft others, while supporting yet more - all on a different subject. Sad to say, in my experience, legislative change was more often driven by a particular constituent of particular importance to the legislator, or a high profile public event, rather than a clear understanding of the effect of current legislation and a desire to alter policy for a public good.
Assuming you haven't funds to "encourage" your legislators to look fondly upon your situation, your best bet is a long, slow process. I'd start by observing that your current ordinance treats all livestock and poultry the same. That's a rarity. Most of the similar legislation I've seen elsewhere treats livestock the same, then carves out a section for poultry. Your proposals legislatively, should seek a similar compromise. Leave the law largely as it is, carve a hole for poultry, supported by findings. That birds are flock creatures needing greater densities to by happy/healthy, that the USDA and others speak in terms of Animal Units Equivalents on land, and that one cow is roughly equal to a handful of goats (varies by breed),or a much larger number of chicken (use your state's resources for the chart - my links are all from FL where I live, or TX, where I used to), observe the history of animal keeping in the area to the extent you can research it, and maybe make reference to the current pandemic and past times when Americans have, in part, relied upon what they raised on their own property to help us thru crisis (i.e. "Victory Gardens").
Reach out via Facebook, others, see if you don't have neighbors also interested and supportive to magnify your voice. and try and figure out who put the existing ordinance on the books, and why - if its no longer relevant, you have another argument for why a modification might be in order.
but ultimately, know that you are lobbying, and its not fast, easy, or cheap.