Is background independence necessary for loop quantum gravity in itself, or only necessary to make LQG compatible with special relativity?

At first look the idea of particulate space in loop quantum gravity would seem to indicate a preferred frame or ether

Not at all – if you're saying this, you're probably not familiar with the mathematics of LQG. LQG is background-independent pretty much by construction. Also, background independence doesn't just mean that there's no preferred inertial frame of reference aka ether, the idea goes way beyond that (you'll need to study General Relativity and/or LQG to understand the physical implications of background independence).

Have they done this in order to prove that LQG is compatible with special relativity, or is background independence necessary for LQG to function in itself?

Special Relativity is a special case, in fact, a solution of the e.o.m. of General Relativity. The specific solution that corresponds to Special Relativity is called the Minkowski space.

The Minkowski space is a special kind of background. GR is background independent, but SR is not: it is based on a specific solution of GR. This scenario is encountered a lot throughout physics, it is known as "symmetry breaking".

LQG is a quantization of General Relativity, so in the classical limit it reproduces General Relativity, and hence also Special Relativity.