Maximise collateral damage by creating an earthquake
The military aren't entirely stupid. They've recognised that just dropping a nuke on Godzilla isn't going to work, and even a succession of nukes might not do it. So in the face of opposition which their weapons can't hurt, they have to go lateral. And with the extinction-level threat that Godzilla represents, there's basically no upper limit on collateral damage that can't be justified.
The West Coast is famously full of tectonic fault lines. A solid earthquake has more energy than any nuke, and (as previously noted in another answer) the Mount St Helens eruption put out more energy than the entire US nuclear arsenal. So some modern-day Barnes Wallis suggests, "Can we blow through the crust, drop the kaiju in the hole, and have it crushed to death in the resulting earthquake?" (Top tip, by the way: Never underestimate the ability of the Brits to put a fuck-ton of explosives to good use. We may talk fancy, but we're a nation of complete nutjobs.)
This isn't even hypothetical, by the way. US underground nuclear testing has already triggered faultline movement, ironically enough in a test called Faultless. So we absolutely know it can be done. The only question is exactly what size of bomb you need to make it happen, which thankfully is something no country has experimented with. Yet.
Enter the nuclear bunker buster. There are still plenty in the US stores, and I wouldn't bet against some researchers still working on new versions somewhere in real life, never mind for fictional plot purposes.
For best results, it seems likely that they wouldn't actually aim at Godzilla. Instead they'd land multiple nukes along the faultline, fuzed to detonate at exactly the same time (and with modern electronics, "exactly the same time" can be accurate to the microsecond). You naturally want a delayed fuze anyway to explode underground. And you just make sure Godzilla is in the middle of the faultline when you do it.
The outcome? Basically a really, really big hole.
The hole truth
How big? In Britain, the bombing range used to test Tallboy bombs, the original Barnes Wallis bunker-busters, still exists and is part of a National Park. (Yes, the village next door is called Sandy Balls. Also don't underestimate the ability of the Brits when it comes to comedy names for places.) From a 5-ton bomb, the crater was 130ft in diameter and 70ft deep. The W88 nuclear bunker-buster for comparison is roughly 95,000 times bigger at 475 kilotons. Let's assume the 5-ton Tallboy produced a 65ft-radius hemispherical hole, and assume volume "excavated" scales with payload - that gives you 3000ft-radius holes. 6000ft wide, 3000ft deep. Let's round down and say the hole is a mile wide and a half-mile deep. According to Google Maps, Owens Valley is around 4 miles wide. 4 bunker busters in a row would then basically guarantee you a hole the width of the valley. And half a mile deep, of course.
How many W88s do we have? Wikipedia says "~400". Let's assume 200 for the OP's question. Suitably spaced, that gets us a hole 4 miles wide, 50 miles long, and half a mile deep. That's probably going to kick off some very interesting tectonic effects, for sure. And Godzilla is going to be one severely pissed off kaiju.
About that collateral damage...
It also gives you 54.6 billion cubic feet of soil thrown into the air per bomb. For 200 bombs, that's 22 trillion cubic feet, or 74 cubic miles. For comparison, Krakatoa only managed to spit out around 12 cubic miles of volcanic ash, and it did significant damage to the world's agriculture for several years. That soil is also going to be highly radioactive - I wouldn't even like to guess at the outcome there, considering Chernobyl put relatively little fallout into the air with rather significant long-term effects.
The nuclear fallout is only the start though. The whole idea of this is to create tectonic effects. If we suppose that this does actually result in an man-made volcano, the end result could be even worse - and what gets thrown up by the volcano isn't just going to be volcanic dust, it's going to be heavily-irradiated volcanic dust. You can most likely say goodbye to California for human habitation for quite some time; but it could quite easily scale up to On The Beach levels of worse. In short, pretty much any level of outcome that fits with your plot can be believably thrown into this scenario, up to and including the end of humanity.
And that's before you remember that you still have to deal with Godzilla.
(Edit: I'd managed to misquote the W88 as 475 tons, not 475 kilotons, when I did my sums. With the correct value, this becomes truly apocalyptic.)