I'm not sure I know where I fall on this. I'm assuming football games are part of your contract/stipend, and that you knew that going in. If you couldn't come teach school during the day because of child care, the district wouldn't keep you on. One of you would have to resign or you would find daycare (I assume the kid is in daycare during the day now). You wouldn't be allowed to have your kid in your classroom all day while you're teaching. Friday night is part of your workday, why should that be different?
Strictly/legally speaking, the district (and your head band director) is within their rights to take a hard line and say you must be at all events (assuming that's part of your contract). Finding child care is your responsibility, not theirs. If you can't do it, you have to quit. They COULD, and I don't see any legal recourse for you if they do that.
Assuming you're both doing a good job and they want to keep you around for the long term, I think a good human response would be to work around it. Find a way that one of you can stay home on Friday nights, especially on away games. The district might want you to give up some/all of your stipend for that. If they do make you give up your stipend, you leave at the last bell every day. Be careful about setting a precedent there, make sure everyone knows this is just for a year or two and you expect that stipend back when you're able to do evenings again.
Taking the kid along with is pretty much a last resort. There are times that my wife (teacher in another district) has an evening event at the same time I do, and we have a frank conversation about which of us could most easily take the kid along. Sometimes we get a babysitter. If you're taking care of your kid, maybe you're physically present at the event, but you're not really being that helpful (especially with a toddler). I have a great photo of me conducting in the stands with my kid on my shoulders, and it's cute, but that night was REALLY hard on me and the kid was melting down by the end of the first quarter. I was NOT an effective band director that day.
One big consideration - how long will this be an issue? You can't leave a kid home alone on a Friday night until they're MUCH older. At some point you can start planning sleepovers with the neighbors, but that becomes an imposition really quick. Do you have any family nearby who could plan to keep the kid on Friday nights for the next several years? If I were your supervisor, and you told me this was for one year and you had a plan for next year, I would probably be pretty accommodating. If you told me this was for the next 10 years.....that's different.