They will still try to decontaminate you of any radioactive materials they can scrub off as a matter of course, but 300 counts per minute, while noticeably higher than background radiation levels, is pretty benign in the grand scheme of things. The fact that you can still count individual radioactive emissions is incredibly good news compared to how bad things could be.
Especially since the reactor will have been shutdown for some time by definition, if the reactor cavity is open enough to fall into. Hopefully the low rate of radioactivity evidenced by the counts on the person's hair is matched by the level of radioactivity in the water.
And on that note, medical attention would also be provided as a matter of course after a fall like this, but it seems to me that the physical injury of falling some distance and possibly hitting metal on the way down is going to be more of a danger than the radiation, especially compared to the sources of radiation people naturally run into (especially cigarette smoke, whether primary or secondhand).
The NRC would make you attend training and get decontaminated if you had to cross a street if they operated the roads.
This sort of place is safe enough to bring your kid into without significant precautions (I got to do this as a kid—it was really cool). The biggest risk by far is drowning.
Relevant XKCD: https://what-if.xkcd.com/29/
0) If you've not read this chart, do carefully read it: <https://xkcd.com/radiation/>. If you've read it before, take some time to carefully re-read it.
1) The guy's getting sent off to seek non-emergency medical attention. I bet you an entire American Nickle that that attention is almost entirely for injuries sustained in the fall, rather than for radiation exposure.
Or to put it another way, 300 CPM (which is a rate) is less than how much radiation you get when on a flight, or how much radiation you get at higher elevations. Even giving a simple explanation of how to calculate Greys (the actual measure you are looking for) takes up the better part of a page. Hell, your bones are radioactive. Yet there are plenty of people posting that somehow the risk to this guy is radioactivity. In reality, his biggest problem is probably going to be finding a new job.
TL;DR you're always getting some ionizing radiation, how much matters.
Are you sure about that? 6200 mSv is 6.2 Sv, which I understand to be near-universally deadly. That dosage would be profoundly incompatible with the news that the worker was being sent offsite to seek non-emergency medical attention.
Poking around, it looks like "counts per minute" have to get converted to a dosage using an instrument-specific formula. I CBA to go find that formula, but you're quite welcome to.