Let's see what they need: a mage who would be able to go to the last level of the Twilight. That would require, as they know, immense raw power, possibly only available to Absolute mages.
The only Absolute mage living at the moment is Anton's daughter, Nadya. However, precisely because she is an Absolute mage, she is the most protected person on Earth, with all three major magical parties having their own protection squad. Even if the trio could get past that, an almost insurmountable obstacle appears: Svetlana, likely the next most powerful mage alive. While there's no way to gauge this objectively, from scraps of conversations one could assume she'd be able to stop even two Higher mages and a Higher vampire, especially for her daughter. Recruiting Svetlana is not an option for precisely the same reason.
Anton, on the other hand, is a (relatively) good choice for a few reasons. He is a Higher mage, so he possesses the raw potential, while at the same time he's still inexperienced enough to be overpowered if necessary. He is useful also because he's been investigating the Crown for a time now, and, knowing Anton for the smart guy he is, the Trio thought he'd be able if not to go fetch the thing for them, then to solve Merlin's riddle and find another, easier way to the last level.
Anton is also easy to manipulate - while he notes that he could've helped them out of sympathy had they approached him, the Final Watch decided to instead blackmail him, knowing he would do anything for his family if the threat was real.
Tl;DR: Anton is an acquaintance who possesses the necessary power, knowledge, and motivation to help them, a combination not found in any other mage they could think of.