It's been about two years since I regularly interviewed people (and longer since I was interviewed), but imagine some of the following might still be helpful.
Presumably, you're interviewing people you want to work with on your team, which is not just about technical skill, but also trust, the ability to communicate, etc.
Repeatedly ask candidates to explain what they're doing and thinking. Rote reading off some text in another window or copy-pasting should be a dead giveaway you don't want to work with the person, particularly if you've asked them not to use AI tools.
Make your interview problem open-ended enough that you can calibrate for somewhat varying ability / nervousness / surprises. If your workplace allows use of AI tools (I imagine the list of those that don't is shrinking rapidly), consider allowing them during the interview but ask them to disclose their use, just as you would expect your colleagues to do while working with you. Someone using, say, Copilot should be able to explain why they are accepting or rejecting the tool's suggestions and show they can adapt the results to suit, on the fly.
If I were a candidate in your interview, asking for anything intrusive such as a phone camera pointed at the keyboard, etc. would likely be an instant deal-breaker for me. Remember that they are (or should be) interviewing you as well.
In my last few jobs we had interview segments which were purely hands-on coding, and other segments which were more design oriented and conversational. I imagine (and now I'm thinking more broadly than your company in particular) that as the tools get better, you'll see more places allowing AI tools for hands-on work, but an increased emphasis on communicating and problem solving. These design problems can be hard to come up with and facilitate, but they can also tell you a lot about how the candidate will communicate, and think on the fly.
Good luck!