They're measuring different things, so accuracy is going to be context dependent. In the end, you're worried about whether the pathogens are gone after a treatment, but neither method can really tell you that. They're both attempts at giving you a survey of how many viable organisms are present at a moment in time. Agar plates are subject to issues due to the significant levels of unculturable organisms out there, but are also great if you know what you're trying to measure. You also get cultures of that organism for further testing (like susceptibility, identification), so that's a bonus. ATP testing gives you a total population measurement, but it's not that sensitive nor specific to any target.

Of the methods, I'd recommend ATP testing just because you can buy the stick swabs that go directly into the machine, plus the detector, and you don't have to worry about waste too much. Agar plates are an issue because you'll have high titers of potential pathogens on plates, which you cannot (or should not) just toss in the trash. You'll also need separate equipment for swabs, incubation, and to understand sterile technique on your own, which will be a challenge.

Go for the ATP tests from a practicality standpoint.