No there’s nothing wrong with sit-ups, you can just easily bruise your tailbone/spine if you’re trying them on a hard surface rather than something padded. If you just want to tighten your obliques (the side bits) you should work on rotational exercises though. I’d suggest using your kettlebell to do Russian twists. Also probably bicycle crunches where, from lying on your back with hands on your head, you bring your knee to your chest and then twist your opposite elbow over to touch it. There’s less of that range of motion that puts pressure on your lower spine like in a sit-up because you’re only crunching the upper abs, but it will still work the obliques really well

Here’s a fitness blog that I actually find useful and less bro-sciencey than most fitness sites. They make a great point that ab routines aren’t really necessary if you’re doing heavy compound lifts like squats and deadlifts as they work your core about as well as a dedicated core workout, but it doesn’t sound like that’s applicable here so maybe just grab some of the exercises they discuss in the post.