While researching a bit about the usage of bayonets throughout their history, I've found multiple accounts that show that bayonet charges were mostly just used as a last-ditch effort or scare tactic. As far as I can tell, there were never any actual bayonet battles. In the early days, one side would affix bayonets, charge (at the cost of 2 or 3 volleys) and, by all accounts I can find, the opposing side would run or die.
Are there any known fights in history where there were actual bayonet-on-bayonet battles?
During the Battle of Borodino, the Russians set up several heavy fortifications on the Radevsky redoubt in order to force the French into an attritional battle rather than a battle of movement that was Napoleon's style of combat. In order to take each of the three redoubts, the French would have to focus artillery and charge with the bayonet because the earthen works would be enough to defend against the French firepower. Further, the Russians did fight in these positions because Russian tactics were more about being stubborn than actually having a tactical doctrine.
However most of the time a bayonet charge is exactly what you mention, basically a scare tactic in order to force an enemy out of a position.
The short answer is: not really.
Mainly because as you noted, bayonet charges were last ditch efforts or scare tactics. And they were awfully effective at being scary. However during the American Civil War there was a certain... enthusiasm for the bayonet and charging with it. One Federal Colonel actually wrote in his after action report that he had intended to charge with the bayonet but his regiment, after being asked twice, refused to do so.
Though there were actions where bayonet charges did occur. For instance at First Manassas when the 33rd Virginia charged an exposed artillery battery with bayonets, they weren't out of ammunition, they just attacked. Or at Winchester, where Jackson ordered the 33rd again to charge an artillery battery if the Federals brought one up, this time the Colonel protested and Jackson told him to "Take it". Hancock at Gettysburg ordered a bayonet charge of the 1st Minnesota when he sent them into the gap in the Federal line, though this one probably fits into the last-ditch/desperation category.
It seems to me from my readings on the subject because it interested me as well, that it's noted most commonly that when one side actively prepared for a "bayonet battle" as it were, by the time they got there the enemy had retreated, due to the scare factor.
The bayonet developed to replace the pike. The pike, and the primary use of the bayonet in the musket era, was to make a formed infantry unit 'spiky' to deter close assault by other infantry or cavalry, sort of what barbed wire might do in more recent times. It wasn't developed or intended as a 'fighting' or melee weapon. The real use of firearms was to kill at a distance, albeit one so short that the bayonet was certainly necessary.
The other reason the bayonet was useful (and isn't much now) was the slow rate of fire of muzzle-loading firearms. If the range of the gun is 100-200 metres, and your rate of fire is once or twice a minute, it stands to reason that you are sometimes going to have to defend yourself in hand-to-hand combat, or, alternatively, rush your enemy over that last gap.
Another mitigating point is that once ordered into attack in this way, an infantry unit would be broken up and beyond control, until reformed. No officer wants this situation during a battle, and it's one of the main reasons why cavalry were so rarely effective in pitched battles, lack of ability to control and re-form the unit.
In these scenarios, you can see that the bayonet is just an accessory, an adjunct weapon to be used against an enemy already broken or bewildered by other weapons. So while there are bayonet charges, any officer asking his men to undertake a bayonet battle was clearly incompetent.