So you cut them a cheque and bite the bullet. If there's one thing Jeopardy! is known for, it's "Think!".
Is this officially being made by Sony or Infinity? If it's the latter, it could come down to licensing costs. I don't know much about Infinity, but I wouldn't be surprised if the rights are more than they want to spend right now. If you really want the authentic experience, you could always do a soundalike think cue, or have someone play the think music from his or her phone.
Case in point: there was a movie called
Hall Pass that used the "DUN DUN!" from
Law and Order. It cost the producers $150K to use a gavel thump. I imagine that cue cost even more.
/Not a movie I recommend FWIW