Really wanted to like Shin Godzilla but felt the enormous majority of the movie was tiresome bureaucratic faff.
I can understand that, definitely. I'm sure you're aware that the bureaucracy stuff is satirical criticism of the government's response to the tsunami and Fukushima, but it doesn't necessarily become funny even once you get that. It's more despairing, I felt, which fits/creates the tone of the film. I love the various mutated versions, feels like he's a real nuclear cancer.
The other one I'd really recommend (beyond the original, which is the best by a long way) is
Godzilla, Mothra and King Ghidorah: Giant Monsters All-Out Attack aka
GMK. It's a direct follow-up to the original and completely ignores the rest of the series. Written and directed by the guy who did the very well-regarded Heisei Gamera trilogy, which basically got him this gig. It's got a darker millennial vibe, the lead actress is likeable and the human stuff is quite engaging. It also takes great delight in killing the annoying characters.
Best of all, once it kicks off, it properly kicks off.
TOTAL FUCKDOWN