Yeah, the 300 can be a bit long but I have found a use for it there from time to time. The water voles can be shy, so standing back and using the 300 can be handy. I also found it handy for the wildcats with the kittens last time, although I was shooting from outside the enclosure on that occasion. I presume you will be inside with them on the photo day and 300 is definitely too long when you are in with them.
Most of what you do on the special access days you will be fine with a 70-200 and may want something a little wider at times too. I carried one body with a 24-105L and the other with the 100-400L at the "official" photo day I did. I rarely went above 300mm, but there were opportunities for close up portraits of the animals where a long lens was really useful.
So, you won't "need" the 300, there are plenty of photo opportunities without it, but you will probably find a few occasions where it will prove useful. It depends on whether you can bothered to carry it. You should get some great images either way.
I would take the fastest lens you can get your hands on, as some of the smaller animals have indoor runs / enclosures which are in pretty low lighting.