It should be noted that that rulebook has been superseded by the
Chronicles of Darkness: Revised Storytelling System Rulebook, which is the most up to date ruleset.
The actual supported Battletech TTRPG game is the Mechwarrior RPG. The original RPG by FASA is simply known as the Mechwarrior RPG and there are three editions:
MechWarrior: The BattleTech Role Playing Game,
MechWarrior: The BattleTech Role Playing Game, Second Edition, and
MechWarrior, Third Edition. Some quirks to note about the earlier editions: the First edition only goes up to the Third Succession War era while the Second Edition introduced the clans. Both of the first two editions outright expected you to be familiar with the rules of whatever the (then) most current edition of Battletech was if you wanted to do large battles and use mechs, which meant that you needed to buy yet more books and be familiar with multiple rulesets, and the First Edition was meant to be compatible with Battletroops for squad based combat. The first edition was almost entirely geared towards creating mechwarriors, in fact, to the detriment of every other potential character type and playstyle. The second edition was also criticized for allowing you to create elite mechwarriors without using any experience points. The third edition, the last edition FASA made, was an attempt to rectify all of these issues; it explicitly did not require you to be familiar with Battletech to play, you could make more than just mechwarriors, and mechwarriors could no longer be made without experience points. It used a character creation system that was identical to the one FASA original Star Trek roleplaying game employed. It was explicitly designed to fix many of Second Edition's issues. The main criticism is that its character creation is somewhat random, along with some apparent balancing issues.
After FASA lost the rights to WizKids, the latter company farmed out the rights to publish Battletech books to FanPro. It was FanPro who published the Classic Battletech RPG. This rpg is not a new edition of Mechwarrior, but is just a reprint of Third Edition, renamed so as not to be confused with WizKids other game line, Mechwarrior: Dark Age. The Battletech game itself was also renamed to Classic Battletech. The Classic Battletech RPG is basically interchangeable with Third Edition, with only the errata being different.
When Catalyst Game Labs (CGL) took over, they effectively rebooted the entire Battletech license with their
Total Warfare rulebook. After that book, they released a series of core rulebooks to go along with it, one of which is
A Time of War, which is essentially Mechwarrior, Fourth Edition under a new name (some fans still call it that). Its the newest version, with the newest ruleset and covering all eras up till the Jihad era (it doesn't technically cover the Dark Ages in the corebook, though that era is mentioned, and it shouldn't be hard to run a game in that era). A Time of War is extremely complex to run (character creation alone is right hard mess to follow), but it doesn't need you to be familiar with Battletech to run. you are probably better following
@Corn Flakes advice and just running your own system and using the books to set the setting and design ships and mechs and such.