Some poking around revealed an astable multivibrator using now-obsolescent ZVNL110A MOSFET transistors. The key idea seems to be large gate resistors putting the DC operating point exactly at the voltage required to hold each transistor in the linear region, pretty much guaranteeing the astable will eventually start up.
A bit of simulation suggests this variation ought to work:
Well, after the kickstarter in the lower left shorts the transistor for millisecond to enforce some asymmetry, whereoupon the simulation ticks along just fine.
The yellow trace shows the voltage across C2 ramping back and forth between ±1.3 V, with a period just over 4 s and almost exactly a 50% duty cycle: much better than the bipolar version, with sensible component values. As before, the cap sees both polarities, so an electrolytic cap isn’t appropriate.
The red trace is the drain voltage at M2 (presumably, “M for MOSFET”, rather than a plebeian “Q” or “T”), which is firmly at 0 V when it’s ON and ramps upward as R4 pulls C1 higher to turn it even more firmly OFF.
The green trace shows the LED current pulse when M2 turns ON at the end of each cycle. Rather than contort the astable into a very low duty cycle, I generate the pulse by dumping current through a smallish cap into the gate of M4. A few tens of milliseconds makes a perfectly serviceable blink and keeps the average current drain down around a milliamp or so.
In between, M3 buffers the astable’s output to deliver enough current to C4. Without the buffer, the cap draws enough current to mess with the oscillations; that’s how I got backed into this corner.
Figuring the LED at 20 mA for 50 ms, the astable at 10 µA, and the buffer at half of 40 µA, the average current of 1 mA comes entirely from the LED, so even a weak lithium camera battery should last a good long while.
If the low average drain ekes 1000 mA·h from the battery, the LED should blink for a month or two before the battery shuts down.