I was over-complicating. It's the same equation as for a geostationary space elevator. In fact, the geostationary space elevator is just one specific case of a general equation for taper ratio. As a mental experiment, pretend that the space elevator equations were applied for a planet of the same mass but with a different rotational period. Clearly the geostationary point would lie at a different point, and you could pretend that such an imaginary planet exists for every non-geostationary orbit.
Borrowing from a previous answer, you could do this simply as follows. The one thing we're introducing is a new criteria for the neutral circular orbit based on the radius, r0, at the middle of the partial space elevator.
$$ U(r) = - \frac{ GM}{r } - \frac{1}{2} \omega^2 r^2 \\
\frac{ GM}{r_0^2} = \omega^2 r_0 \\
U(r) = - \frac{ GM}{r } - \frac{1}{2} \frac{ GM}{r_0^3 } r^2 \\
\frac{ \lambda(r_{0}) }{ \lambda(r) } = \exp{ \left( \frac{ U(r_{0}) - U(r) } { \left( \frac{ \sigma}{ \rho} \right) } \right) } $$
The two independent variables are r0 and r. The final expression is the taper ratio. You need to know the free-floating central point and you need to know how far it extends in either direction. Yes, it does need to be balanced on both ends, and that would be the more difficult requirement. In order to find the length on the "other side" that corresponded to a radius selection you would see to set the integral of the linear mass-thickness on both sides to be equal.
