The KT family or rockets was allegedly derived from the DF-31 programme. This LM 11 is claimed to have a smaller diameter than those. So it may be a custom design for space delivery, rather than something having to do with ICBM. After all, there's also the DF-41 with similar/even larger diameter to DF-31.
JL-2 is allegedly 2m in diameter, similar to LM 11. But one would expect the Chinese to go with slightly larger diameter, closer to Trident, for their JL-3 missile.
Solid propellant has been used for space launch before so it's not an indicator in itself...
On another not so related note, how similar could deployment of several satellites be from deployment of several warheads? Technologically speaking.