I wouldn't stress about the lidocaine and what is already done. Some pain meds that end in 'caine' can be toxic, but some dosed appropriately are used by vets. Most here recommend avoiding them to alleviate confusion and possibly using one that shouldn't be used and doing harm. Many times I do not use pain meds since the pain will make them self limit movement and prevent them hurting themselves more, but I understand why you want to help. I'm attaching a couple of links with info on pain treatments, unfortunately many of the commonly used meds have to be gotten from a vet.
I would watch for signs of infection, since the wound was extensive. Did you make sure the crop was intact and not damaged? Asking since I didn't see it mentioned, may have missed it.
If the crop was torn then food ingested could leak into the neck area and increase the risk of infection. I would consider either starting a round of antibiotics, or at least having one on hand in case it's needed. You can get fish mox at some tractor supplies, to give orally.
These are for when you have time to look at them:
https://www.chickenvet.co.uk/news/post/analgesia-in-poultry
https://www.tandfonline.com/doi/pdf/10.1080/17415349.2017.1395304