You need to meet your child at your child's level in order for them to feel happy in their learning environment. Sending this child to school may or may not be a disaster - you are only making assumptions that it will not work. There are a lot of good things about schools - things that get talked down when homeschooling is being touted.
I personally would like to homeschool but in our child's case, the benefits of the school environment far outweigh the benefits of learning at home. We make up for the lack of sufficient acceleration at school by afterschooling intensively and sending DS to a school that will accomodate his learning needs somewhat.
Since you would like to prove that your DS is ahead - you can have him take a placement test with Singapore Math or the EPGY Mathematical Aptitude Test. I remember doing these when my DS was 4 just to get an idea on where he was at.
As for listening comprehension, you can ask your DS to summarize and narrate an advanced book that you read aloud to him and show your DH the video of that narration.
Also, you might get some above age level logic workbooks from Critical Thinking Company or Prufrock press and have your DS complete some of their mind bender like puzzles to show his advanced reasoning skills.
It is hard to quantify the "above grade" performance for a 5 year old in all the other subjects like history, science etc. in my opinion. That will be only possible when they systematically study these subjects at a later age.
