Is an RSA variant with public exponent $e=f+(p-1)\cdot(q-1)$ safe (for $f$ random in some small interval)?
I can't see any problems, if you use it as specified (with the performance advantage of a factor of 2).

If you start to narrow the interval too much, you're going to get into trouble.  For instance, one factoring method is to compute $c=b^e \pmod n$ for some known base $b$; then try to compute the discrete logarithm of $c$ to base $b$.  We know $c=b^f \pmod n$, so if $f$ comes from a small range, so we can use some algorithm for computing the discrete logarithm when the exponent is known to come from a small range (e.g., baby-step giant-step or Pollard's rho).  This will reveal $f$, if $f$ comes from too small of an interval. For instance, if you pick $f$ from the range $3 \le f \le 2^{80}$, these algorithms will be able to find $f$.  Once $f$ is known, $n$ can be factored, as $e-f$ reveals $(p-1)(q-1)$ which can be used to extract a factor of $n$.

Similarly, if you choose $f$ to have very low Hamming weight, I think you will again get into trouble.  The baby-step giant-step and Pollard's rho algorithms can be adopted to solve the discrete log in cases where the exponent is konwn to have low Hamming weight, so the same method can be used.

For these reasons, I don't think you're going to be able to get a very large performance advantage.

And, I have to say that I would not bet much of my own money on the security of this scheme.  I believe van Oorschot and Coppersmith have some results on factoring $n$ given partial knowledge of $d$ or $p$. The lesson I draw from those results is: when you start giving away partial information on RSA secrets, RSA can lose a lot of security.  I don't know if that applies to your scheme, but I would be reluctant to use it in a high-stakes situation, and I would be especially reluctant to narrow the interval from which $f$ is chosen.