Can we stop pretending this isn't about nuclear weapons research and - basically - continued funding?
LLNL is at least a decade behind its initial predictions of ignition. The current lasers may be inefficient but they've been fine-tuned very precisely for optical quality. There's no guarantee more efficient lasers would have the same characteristics and could be focussed in the same way.
I'm sure LLNL know the maximum theoretical fusion gain for a realistic pellet design, and it's worth nothing that that number hasn't been mentioned anywhere. Clearly there's only so much energy available for each cycle. If that energy is on the wrong side of what's needed to release it even after efficiency improvements, the entire system isn't workable, no matter how it's re-engineered.
The laser practicality issue that prevents this from directly becoming a power source would also be a major barrier to its application as a weapon. The laser fires in the UV-B (351 nm), which is scattered and attenuated by air, to say nothing of smoke or dust; it also requires incredibly high targeting precision (<2 mm target diameter) and consequently precise placement of a target weighing only milligrams. Additionally, the various optical components of the laser must be very well aligned, which is difficult to achieve in any battlefield conditions. And the whole thing is a very obvious and vulnerable target.
I ascribe a small possibility to its utility as a weapon in space, but practically zero on Earth without other major developments.
>I'm sure LLNL know the maximum theoretical fusion gain for a realistic pellet design, and it's worth nothing that that number hasn't been mentioned anywhere.
I wouldn't be so sure. Fusion is in general quantum chromodynamics, which is not so well characterized (being the subject of the famous YM mass gap conjecture). Even in this case it was stated that the yield exceeded expectations and damaged the sensors, which was probably not desired.
That is not true. You can of course not make a 10 % efficient laser more than 10 times more efficient, but there is no limit to how much more efficient you can make a laser in general, you just have to start with an inefficient enough one.
And the NIF lasers are quite inefficient, according to Wikipedia - which might have slightly outdated numbers as someone pointed out to me - they turn 422 MJ stored in capacitors into 1.8 MJ UV laser light, that is an efficiency of 0.43 %. Bringing this up to the 20 % you mentioned would already almost be a factor of 50. Also not all of the gain has to come from more efficient lasers, the fusion process could also be made more efficient.
But you are of course correct that there is some general limit to laser efficiency, therefore the question is if the fusion gain can compensate this and all additional losses during the conversion into electricity further down the line.