Mulan is one of the first Disney animated movies to have anything to say other than "Friends are good" or "Beauty is on the inside" or "Someday you'll have to let your 16 year-old mermaid daughter leave forever with a handsome stranger she just met." The central conceit of a girl masquerading as a man to go to war in the place of her ailing father was relatively progressive for a "Disney Princess" movie. Too bad it doesn't make any sense.
If you remember, a lot of the tension in the story stems from the potential consequences if Mulan's true identity is revealed to the Chinese army. The movie tells us several times that the punishment for the crime of Reverse Mrs. Doubtfiring is death. This is the reason Mulan's father gives for not running after his daughter; to expose her would be tantamount to executing her himself. We don't see Papa Fa Zhou again until after Mulan returns home with a big shiny medal. He's just happy to have his daughter back.
But there's a problem: Everyone saw Mulan's dad accept the order to go to war.
Mulan's family is pretty well-known in the community. When Shang first hears the name of "Ping's" father, he says "THE Fa Zhou??" like a kid who sleeps with a Fa Zhou action figure. There were tons of people in the square that day when the notice was handed down, and after Mulan made a scene about it, her attempted meddling was probably the talk of the town.
Which means that if anyone who saw Mulan's father in the weeks/months since her departure, the game would be up. All it would take is for Fa Zhou to stroll into the market, and a half-dozen people would say "Hey, wasn't that highly-decorated local celebrity soldier conscripted a while back?" Word probably would have gotten back to the army eventually, which would mean a grim end to this fairy tale.
Mulan technically has a guardian dragon for these kinds of occasions, but he's not exactly invested in saving her.
As far as Disney sidekicks go, Mushu is lacking something compared to the likes of Genie or Timon and Pumbaa. He provides some much-need comic relief for sure, and really does save the day at the end of the movie. But for 85% of the running time, Mushu is only out for himself. All Mushu wants is to regain the respect of the elders and become a guardian again, and he'll do anything to attain that goal -- up to and including putting Mulan in harm's way.
Technically, Mushu's job was to wake up the bigger, better dragon from its slumber so that it could go retrieve Mulan. It was only after Mushu accidentally destroyed the "real" dragon's statue that he decided that the only way to get in the good graces with the family ghosts was to make Mulan a war hero. Basically, Mushu was willing to sacrifice Mulan to get his old job back.
Mushu actively tries to get Mulan into fighting shape not because he really likes her, but because that's simply what's going to help him most. This is exactly the same kind of attitude that got Mushu demoted from guardianship in the first place. We briefly see his last master, who was beheaded because of Mushu's screwups.
It's clear that Mushu is capable of leading people into danger, but for most of the movie he shows no remorse for doing so. Hell, when the wormy Chi Fu decides to spite Shang's group by keeping them out of the war, Mushu takes it upon himself to forge orders so that Mulan can get into battle.
Mushu deliberately decieves Chi Fu (who, to be fair, is being a total dick), in order to send Mulan off to the front lines. "I worked way too hard to get Mulan into this war," says Mushu, not at all concerned with the reality of mortal combat with a brutal and unyielding enemy. Emerging victorious will honor the family, but to Mushu that's merely a means to an end. He's out to get his, Mulan's safety bedamned.
All of this is resolved in a couple sentences after Ping is revealed to be Mulan. Though Mushu apologizes for being selfish, it's quickly brushed away. Mushu never faces the concequences of his actions, and no one is ever really angry with him for purposely risking the life of the one person he was supposed to protect. Overall, Donkey from Shrek was a much better Mushu than Mushu.