It seems more likely that scientists used the weapons angle to dip into that massive flow of military money for their energy program.
But, ICF is simply not a viable way to produce fusion power, it is far far far too expensive to operate such a device. So, we can only conclude that they are either deluding themselves, or they are in fact doing fusion weapons research (or, at best, simply fundamental theoretical research into how fusion works) - since the same kinds of conditions or forces are what happens inside a fusion bomb.
Note that in the US nuclear weapons are controlled by the DOE, and not the DOD.
It's arguable how much of the NIL is still validating models useful for simulating nuclear weapons vs fundamental research vs energy research. And it’s likely the relative importance of each factor has changed over the programs lifetime.
You can argue all you want. Lawrence Livermore National Laboratory is a government-owned, contractor-operated facility managed through a contract between the LLNS Board of Governors and the DOE's National Nuclear Security Administration (NNSA). The NNSA in turn works to ensure that the nation's stockpile of nuclear weapons is safe and secure.
The funding source being mandated solely for nuclear weapons goes against your claim it's not primarily for weapons.