This is really good. The story was very entertaining and hilarious at times. :)) I really like the plot and the characters, you are lucky to have such a gaming group. Please keep this up. Not only is a great read but very helpful for my own games.
I like very much the "Social General" theme of the story. Though I think the myth of Lyran military incompetence has been vastly exagerated, it helps explain why a a regiment can be rated as Green, and on the other hand, neither the Lyrans nor any other Inner Sphere faction are as professional as the Prussian General Staff. I like very much that in keeping with the setting military professionalism often takes a backseat to politics or noble birth. Generally speaking, the Great Houses armies are very much on par regarding military skill, or lack of it, and success is more having to do with superior forces than finesse.
Some themes I think you should address. First of all, Lyrans love money. You should show how corruption, mismanagement of funds, and war profiteering are rampant. Since Lyrans are businessmen at heart, you should stress how many individuals see the military as just another business venture.
Second, you should show the scavenger nature of the Successor States and that Mad Max post apocalypse survival, technobarbarian theme, with failing and scarce equipment, and the contrasts between advanced technology from yesterday being undistinguishable from magic and treated with reverence, and in some cases technology in Battletech being so backward for things that today we take for granted. This could nicely tie in with the corruption theme, hoarding spare parts for private use is a court martial offense, but done anyway. I think your players need to be remembered that money is not everything, Mechs are. That their elevated social status is so dependent on their machines, and how fragile and precarious their position actually is since it rests on a fragile, mended, prone to breakdowns centuries old piece of machinery.
Third, finally, this is a war game, unless you and your players enjoy the peaceful and boring garrison life routine, players should be reminded that this is a time of war and their characters are going to get into harm's way sooner or later. It is all fun and games until someone gets shot. The Inner Sphere is not a happy place, is really pretty much a crap sack universe. Is like the Twilight 2000 setting, centuries after the Third World War... but not just on Terra, everywhere. The motto of the game is "Life is cheap, machines are expensive". The Lyran military in particular at times has overtones of WWII Red Army style brutality, and is not just the massed numbers and the firepower over subtlety, if you read carefully the sourcebook, the Lyrans sometimes resort in the name of expediency to brutality that would even make the Draconians balk.
I am not dictacting you what to do with your game, just offering some suggestions of what I believe are the fundamental themes of Battletech and wich I thing they should be included at one time or another, to a major or lesser degree in Battletech games. I left out politics, because I think they are just of average importance in a Steiner or Davion unit, compared to the extremes of unquestioning obedience and toeing the party line of Kurita and Liao, and the factionalism and infighting of Marik.
So summing it up, ideas you might find useful for your campaign:
1) Money, money, money, makes the world go round.
2) Pray to the Bomb, err, the Mech. Or, my kingdom for a spare gyro.
3) Life is cheap, machines are expensive. Lyran variant: Don't bother about collateral damage