How would you propose he catch her cheating, including proof beyond reasonable doubt, that would guarantee a similar settlement or better?

If there is a start recording button there is a stop recording button, he could have tried to avoid filming during sex perhaps. Maybe there are other camera's in the house, or a video doorbell.. but generally I agree on this: it will be his word against her without any proof, the proof won't be proof when they're just in the same house at the same time. What's also weird if he would have entered the house and caught them having sex he wouldn't have invaded her privacy. If I were him i'd be upset I had to pay money but relieved to be receiving some and if i were her i'd be upset I had to pay money but relieved it could've been worse.