Just looking over Fanbook 1 - Field Manual Terran Republic 2785 once again. There are several manages of the Gunslinger Program and its reestablishment by the Republic. Here are some mentions of the Advanced Combat Maneuvering School (ACMS) or Gunslinger Training in the book.
PDF page 87
The TRAS has however designated the War Academy of Mars (WAM) the headquarters of the Gunslinger Program which means that the school will train the very best Mechwarriors. Graduates will likely be assigned high profile jobs in the Terran military thanks to the Academy’s continuing reputation.
PDF page 94
It is needless to say that postings in the Terran Guard are highly sought after not only for the prestige of such an assignment but for the additional experience that can be gained from specialized training. This training ranges from Gunslinger courses to advanced strategic operations exercises given at Republican military academies within the Terran Province. Despite the organizations young age officers from the Guard have already gone onto high positions within the TRAS a trend that is likely to continue.
PDF page 102
The Dieron Borderers are primarily composed of the heaviest military vehicles due in large part to their Star League predecessors. Every member of Dragonsbane, as they are nicknamed, has a solid grasp of Kurita tactics and strategy. Of all the Borderers they have the most Gunslinger trained personnel within their units. Their daily training continues to focus on the warriors of the Dragon and the coming conflict with the Draconis Combine. During parade functions, a rarity these days, the Borderers of Dieron use a gold and white base color scheme to decorate their equipment.
PDF page 116
As with the Black Watch all Fusilier Mechwarriors must be a graduate of the Gunslinger training program. This has only recently been accomplished with the reestablishment of the Terran Gunslinger Program.
So for the Terran Republic we already have a reestablished Gunslinger system headquartered at Mars. The Terran Guard make extensive use of the training offered while the Dieron Borderers have the most Gunslinger trained personnel within their units. That is impressive since all three Mech regiments of Sinclair Fusiliers are Gunslingers.
This covers the restoration of the program following Operation Liberation circa 2780. However that still gives Muttley some 13 years to play with for the War Academy of Apollo. Not a bad idea to cover the inter war years. Kerensky liberated the Rim in 2767 so you could start there for the date. However they'd face many challenges. Native opposition from the people of the Rim, Commonwealth invaders, and Draconis Ronin.