Does this not just come out in the wash based on their performance? Like, even if they were say working 6.75 hours instead of the 7.6 standard per day (38 hour standard work week) and were getting their tasks done to a high standard, who cares?
With the time spent not commuting, they're probably hitting their work hours easy.
This is obviously not legal advice, but the angle I'd take if I I were you is if their performance is lacking, just don't even mention the kid at all, it should be squarely focused on performance, that's the only thing that matters to you.
Just my two cents that I think this won't be a big deal in the end, even for employers. Covid has shown us working from home works just fine for most desk jobs.
Disclaimer, I don't have kids, and am an employee of a business where I have to submit timesheets, so these comments are based on my own opinions and not from experience or expertise.

Is this not a type of performance monitoring? And to be fair, would performance not be monitored in the same way (assuming it's a desk job) not matter if they're working from home or not. I just am usually a bit sceptical that WFH is much of a burden.
This might be a little un-generous, but it seems that you believe the employee is making more mistakes at home, but it could also be your own bias that you know they have a child at home, and you think they're more distracted. I'd encourage you to just look at their work and decide from that if there's actually an issue here (there may well be, I obviously have no idea).
You'll save yourself headaches down the line if you end up putting them on a performance management plan (if it really is a serious issue and not overreacting, again I have no idea), if you base it entirely on performance, and that way there will be no accusations of bias.
If they're getting their work down, I'd encourage you to not worry about it and if they're not, the issue is with them not holding up their end of the bargain - not WFH.
Me personally, I like being in the office, because it's so much easier to bother each other for a quick chat about XYZ issue. When you're at home all you can do is message, since it's hard to tell how focused someone is. At the office you can base it on vibes. I am also single with no kids, so I am strongly in favour of WFH for everyone by default.
I obviously won't tell you how to run your own business, especially since I've never managed a team before, let alone a small business. But I'm still finding it hard to imagine why it's so much harder for a small business (if it's just a desk job).
Anyway, just my 2 cents to encourage you to catch yourself being biased because of children (if you are, you may be right on the money!)
Good to catch with ya :)