Can I use different IVs when encrypting identical plaintexts with identical keys with AES-CTR?
I'm wondering whether I should try to use the same IV each time,

No. Using the same IV twice for the same message means that you leak that you sent the same message. If your set of messages is somewhat small this allows an adversary to more easily correlate reactions to messages with possible message contents.

whether it's safe to use a random IV for different instances of the same plaintext.

Yes, it is. The security definition which CTR satisfies (CPA security) very much considers the fact that the same message is sent twice. This will not leak anything (within reasonable bounds). To see this note that $F$ is usually modelled as a PRF, that is if your adversary doesn't know the key it might as well be a truly random function $f$ which takes its input, draws a response uniformly at random and stores the mapping for this input. This means that $F(K,I_1)$ can be translated to $f(I_1)$ which can be translated to the random string $R_1$ and the same can be done to construct $R_2$. Then if you look at $C_1=P\oplus R_1$ and $C_2=P\oplus R_2$ you will notice that both are &quot;essentially encrypted with the one-time-pad&quot;, i.e. independent and uniformly random keys of size equal to the message length which will obviously not leak any information.