OK, this was a rushed question. But in case someone might be interested in the same in the future, perhaps a self-answer will be helpful.
Looking at GSL's source code, I found that F(φ,k) is reduced (as long as φ ∈ [-π,π]) to
sin(φ) RF(1-sin²(φ), 1-k²sin²(φ), 1),
where RF is another GSL function. There's no F(iφ,k) but we can safely plug a pure imaginary φ in the above as all the three arguments stay real:
sin(iφ) = i sinh(φ)
=>
F(iφ,k) = sin(iφ) RF(1+sinh²(φ), 1+k²sinh²(φ), 1)
=>
-i F(iφ | -k^2) = sinh(φ) RF(1+sinh²(φ), 1-k²sinh²(φ), 1)