It may have training in modern empire-vs-empire warfare, but it doesn't have any established experience in it.
I wouldnt say thats the case at all. The Vindication Wars were a long running conflict between the Republic and the Ammatar Protectorate, which is still a nominal part of the Empire. I think it makes sense that the Empire sent military advisers or even whole military units to aid their Ammatar vassals and protect a part of the Empire. If anything the Empire likely had a lot more "hands on" experience in modern warfare heading into the Empyrean Age than either the Fed or the State.
Bold is mine. There is little reason to think this is entierly true - The Empire might have "more" experience due to a practically on-going military conflict prior to the new war, but the State and Federation each had their own issues to deal with long before the war started.
Ignoring any covert military action against each other, there are still the Guristas and Serpentis breathing down their necks. There is no reason to think either Federation or State did not jump at any and all chances to smack them down when and where possible. Also, intra-corporate knifing in the dark inside the State and who knows how many local conflicts and wars might have compelled Federal Forces on 'peace and order' missions during 100 years of official Union-wide peace.
So, let's look at this publicly available description of a corporation here...
The Templis Dragonaurs are an ultra-nationalist Caldari terrorist organization whose origins date back to the Tikiona States. They are the most anti-Gallentean political entity in New Eden, and were secretly involved in the destruction of Nouvelle Rouvenor.
They are the most anti-Gallentean political entity in New Eden, and were secretly involved in the destruction of Nouvelle Rouvenor.
secretly involved in the destruction of Nouvelle Rouvenor.
Yeah. Something like that.
I'd say Templis involvement and existence is a publicly known fact by now, as well as their control over the State proper, through Heth.
If not... well, I'll accept any sufficently proper hand-waving to explain how this can be available corp info but still unknown...