Honestly, I think a sniper would come in handy for several reasons.
One, if it does have a small weak spot he can target it more precisely. Besides, it's not just bullets we might decide to put into a kaiju. If our science boys come up with some sort of tracking device or anti-kaiju poison, that sort of thing, he's the man to deliver it.
Two, a sniper is observant and used to working on their own or with a spotter. They might see things others miss. A good use during a kaiju attack on a major city might be to set him down on a rooftop with binoculars and a radio to keep the rest of us informed on where it's going and what it's doing. Valuable intelligence.
Three, we're not always going to be fighting one huge animal- sometimes it might be a number of smaller ones, or even humans.