Where do you live?
This site gets international traffic. The risks of wildlife isn't the same everywhere.
Seen globally, unprovoked wild animal attacks are RARE.
Depending on where you live, this might be upgraded to AMAZINGLY rare.
Can't think of many places where it might be downgraded to a realistic concern. Maybe someplace where polar bears roam.
I tried a search for "cyclist killed by wildlife", but there were no early hits on any useful statistics.
Particularly if one were to add your specifics - "at night" - it doesn't seem to happen often enough to make any reliable assessment of the risk possible.
As an example, Oregon has had ONE fatal cougar attack in 94 years. Looking to your whole life, you're far more likely to die from fall down a flight of stairs than from a cougar attack.
So, unless you live someplace where wildlife is known to get territorial or hungry for humans, worrying about a wild animal attack isn't a sensible thing to do. If you work on a farm, worrying about wildlife isn't a sensible thing to do.
At the off chance that you actually do live someplace where wildlife is a real threat, an axe is a poor choice. It's a short-range weapon. It's cumbersome to draw. It only has a 4-5" cutting edge.
Sure, if you get in one GOOD hit, that will finish the fight with any animal not having an armoured skull/neck.
But this would depend on you realizing you're about to be attacked BEFORE that attack happens.
And most predators are ambush predators.
At night, in the forest, the odds of you noticing the danger before the animal strikes is minimal.
Attach bells to your bike, let wildife know that you're coming. That'll take care of all "normal" animals. For the remainder, an air horn, bear spray, a starter gun/gun firing blanks would be a better option than an axe.
Any of the above that can be mounted to the bars would be my first choice.