Okay, the short answer to your question is that the only way you're likely to stop the other company's use is to file a lawsuit and win. Since we're not sure about the outcome of that suit and since a lawsuit could cost more than buying a house, the smart thing to do would be to wait to see what happens with your trademark application. If your registration is granted, you'll have a better foundation for your lawsuit. The big question you will need to ask yourself before walking into any court is whether the competitor is hurting your business. Are your fans really likely to be confused?
What Trademark Rights Do You Have?
It's possible that your competitor may be waiting to see what happens with your application or may even file papers to oppose the registration. Not all applications are approved. For example, an examiner may object that your trademark describes some aspect of the services (it's considered a non-distinctive or "descriptive" mark). If the examiner objects, you'll need to demonstrate that consumers associate that mark with your services. The examiner will also research other uses of the mark and confusingly similar variations (like BY KIDS ... FOR KIDS) to make sure you have exclusive rights in your field. You're probably aware for example, that PBS uses a similar tag for Zoom. And WBGH owns a federal registration for BY KIDS FOR KIDS for entertainment services (Reg. No. 2691810). There's also a website that uses the acronym BKFK ("By Kids for Kids") and they have a federal registration as well (Reg. No. 3472214). (And there's also a pretty funny parody using the phrase.) We're not sure what all that means for your application, but it's a reminder that getting a registration is not always a sure thing. (You can read more about the trademark application process in this guide prepared by the Dear Rich Staff) If the application is rejected, your competitor may feel more confident about its use. Without the registration you may still have rights and the geographic location of your uses -- locally, or perhaps over the Internet -- may affect the outcome of any court battles.
What Trademark Rights Can You Afford?
Let's say your registration is granted. Great! You will have exclusive rights for broadcasting and streaming over the Internet. But you will also have to take time and money to go after others. And if you win, you may not get back a huge judgment or your attorney fees. That's why we urge you to go slowly. If the competitor doesn't really affect your current or future income and your fans aren't confused, you should probably hold off on doing anything. Whatever you do, don't file the lawsuit just because the competitor makes you mad. As Frank Sinatra once said, "The best revenge is massive success."
PS If you need help understanding any of the legal words used in this answer, check out Nolo's free Law Dictionary iPhone app.