Try adding some series resistance to C3, like an ohm.
Putting a 10uF MLCC capacitor (incredibly low ESR) on the output would make a real life circuit go crazy, too. Adding an ohm in series would bring it closer to an electrolytic capacitor.
Go with NANDBlog's solution, because it makes no sense to add unnecessary parts into your circuit if you can just pick a different part and solve the problem that way. As you probably figured out, I'm not too familiar with LT voltage regulators.
For modern LDOs low-ESR capacitors are absolutely fine (and recommended or even required!), this is mentioned in the datasheet. Surprisingly, MLCC is also fine for plain-old regulators like lm317 (if I'm not wrong) because of their output cascade (common collector or voltage folower, I don't really know what I'm talking about, I'm more into MCUs) does not have voltage gain and, thus, stable with MLCC or event without a capacitor.
Concerning using different part. Besides its slightly worse haracteristics (like PSRR), it also does not have separate supply for built-in circuitry. This greatly reduces dropout. LT3080 has like ~0.35V dropout, LT3081 is around 1.5V. This makes some difference because I've got a pre-reg and can significantly reduce heat dissipation (I'm trying to fit a 50-100W PSU in a relatively small box (just for fun and to save some space on the bench)).
Anyway, both parts work unstable in the simulator under some conditions. But I'll recheck tomorrow, may be I did something wrong (meanwhile, there was also a mistake with voltage double, I fixed it).