The important thing about being anti-racist is to be resilient and open when challenged. And to discuss things. Because we all have different triggers and tolerance levels. So I absolutely would discuss with them. However, I’d speak with your manager first, to make sure it’s OK to talk about the complaint. The client may not have wanted you told about it. You don’t want to make it worse by breaching confidentiality.
Watermelon is a racist trope associated with people of African descent being dirty, lazy and childish. So, if you mentioned this in a way that could have been taken that way, it may have been perceived as veiled racism, e.g. “You’ve done well eating your watermelon today, haven’t you” (e.g. speaking to someone like a child and implying they may be lazy with their eating). This may be helpful: https://www.theatlantic.com/national/archive/2014/12/how-watermelons-became-a-racist-trope/383529/.
The important thing is to create a safe space, where they feel they can talk to you easily and openly. And that you’ll hear them, and take them seriously. I’d say something like: ‘I hope you don’t mind me mentioning this. My manager told me you’d complained about me. I’m sorry I offended you. I’m keen not to do it again. I know references to watermelon can be racist. But I didn’t mean it in that way. Could you explain to me whether it was because I mentioned it at all or if it was the way I said it? I’m really keen to make sure I don’t offend you or anyone else. I’d really appreciate it if you could let me know if I say or do anything else that offends you, as it’s important I know’.
Ignoring the complaint would be the worst thing. Imagine complaining to an all-male company about something someone said which you took as sexist, and then nothing being said? So, if your manager suggests this, I’d challenge them on that and say you feel it’s better to take the complaint seriously, apologise for offence and talk about it.
Also, having mixed ethnicity kids doesn’t mean a person automatically isn’t racist. Black people can say and do racist things, because we live in a racist society. Racism can be indirect as well as direct, e.g. treating a person differently because of racist assumptions (even if well meaning) or organising things that make it difficult for people of colour to participate, e.g. a job interview at Eid.
And people of colour can be racist about other ethnicities, e.g. people of Caribbean-African descent making racist assumptions about, or being directly racist to, people of African descent. Or people of Indian heritage making remarks about people of Pakistani heritage.
Also, if you’re White, while having Mixed kids will have given you insights into what it’s like to be a person of colour and the direct and indirect racism they can experience, it doesn’t mean you know what it’s like. You may have already read it, but I recommend Akala’s book: ‘Natives: Race and Class in the Ruins of Empire’. He talks about this in it as his mum is White.