  Layered perovskite iridates realize a rare class of Mott insulators that are
predicted to be strongly spin-orbit coupled analogues of the parent state of
cuprate high-temperature superconductors. Recent discoveries of pseudogap,
magnetic multipolar ordered and possible $d$-wave superconducting phases in
doped Sr$_2$IrO$_4$ have reinforced this analogy among the single layer
variants. However, unlike the bilayer cuprates, no electronic instabilities
have been reported in the doped bilayer iridate Sr$_3$Ir$_2$O$_7$. Here we show
that Sr$_3$Ir$_2$O$_7$ realizes a weak Mott state with no cuprate analogue by
using ultrafast time-resolved optical reflectivity to uncover an intimate
connection between its insulating gap and antiferromagnetism. However, we
detect a subtle charge density wave-like Fermi surface instability in metallic
electron doped Sr$_3$Ir$_2$O$_7$ at temperatures ($T_{DW}$) close to 200 K via
the coherent oscillations of its collective modes, which is reminiscent of that
observed in cuprates. The absence of any signatures of a new spatial
periodicity below $T_{DW}$ from diffraction, scanning tunneling and
photoemission based probes suggests an unconventional and possibly short-ranged
nature of this density wave order.
