While projected would be understood to imply that it was in the air for everyone to see, a more specifically appropriate word, your sentence aside, is broadcast:
2 : to make widely known
A man who—make no mistake—cares deeply, but doesn't feel the need to broadcast it.
— Allison Glock
3 : to send out or transmit (something, such as a program) by means of radio or television or by streaming over the Internet
// an event being broadcast live on television
The problem with your sentence is that while projected is perfectly fine for the sentence itself, it's not an ideal explicit example of the "put it out there for everyone to see" idea.
If I were to take your sentence and apply broadcast instead, it would become:
The cops broadcasted the bat signal.
Note that onto the clouds doesn't work well after this particular verb. However, you could add something like this:
The cops broadcasted the bat signal for everyone's awareness.
Also note, however, that broadcast still isn't really a good fit in conjunction with the bat signal. If I were really trying to turn this into a good example, I would make it something like this instead:
The cops broadcasted an appeal to Batman for his help.
Such a broadcast would take the form of a projected bat signal.