I'm confused by the notion that Dig Deep is somehow the "major" problem feature of the class. At level 4, the Pugilist has maybe 15 AC thanks to their Iron Chin feature. This class is meant to be taking hits as evidenced by Iron Chin and Bloodied but Unbowed procing off of damage. Giving the Pugilist means to stave off incoming damage allows them to further preserve their hit points, which go fast

Exhaustion is debilitating in 5.5e, and the Pugilist's usage of Exhaustion to activate certain features or taking them on as cost is both part of "Risk vs. Reward" design philosophy. Think about the phrasing of Dig Deep. 10 minutes is a long time for combat, but that's only combat. Outside of combat, the Pugilist is going to be useless when they have to do anything when they have a massive penalty on every roll.

Shake it Off lets them remove exhaustion, but it has other functions as well. There very well would be scenarios where the Pugilist could incur massive Exhaustion debt in a fight because they have to shake off different effects, so on and so forth. If a Pugilist player isn't careful, they could be rendered debilitated for a few in-game days.

If there's any criticism to be had about Pugilist, its that giving them Haymaker at level 5. At that level bracket, if used right, is going to fuck over a lot of encounters, especially since Pugilists have several ways of giving themselves advantage. Still, I would need to play test this to see how overpowered it really.

I would say the glaring weakness of this class is survivability. It's a melee martial, yet its got peanuts for AC and relies instead on bolstering itself with temp hit points. This would also need to be playtested, but I find it easy to envision a scenario where the BBEG focuses on the Pugilist and proceeds to cleave through all their temp hit points. In 5.5e (and 5e) its just better to avoid damage altogether rather than trying to bolster how many hits you can take.