Washuu will probably teriffy Dumbledore for one reason; she's smarter than him. Dumbledore is a genius, and he can probably count his intellectual and magical equals on one hand. It's very unlikely he's met anyone much smarter than him in his entire life. And Washuu will run rings around him easily. He probably won't do anything stupid, but he'll definitely be rattled. The other thing is that secrecy is Dumbledore's most serious flaw. He hordes information, keeps his confidants and associates close to him, and never gives out important or dangerous information if he absolutely doesn't have to. Honestly, he's a lot like Urahara from Bleach. While his compulsive secrecy helps him at times-Dumbledore's espionage tactics run circles around Voldemort's- but it hurts him just as often, and it'll hurt him with his interactions with Tenchi and Co. Long story short, he'll likely make a bad first impression. Even after Dumbledore is convinced that Washuu and the others will be good for Harry, he'll probably try and keep their visibility at Hogwarts to a minimum- not out of desire to separate Harry and his quasi-adoptive family, but simply because he doesn't want people finding out about Harry's relatives. Given the wizarding world's attitude to non-humans and muggles, he may actually have a half-decent political reason for his secrecy.
Dumbledore's other serious problem during the books is that outside Hogwarts, he really lacks any sort of official authority. If I remember correctly, he actively avoided taking positions of official authority because he was scared that he'd succumb to the temptations of power. Unfortunately, unofficial power requires people to listen to you- once's Fudge's paranoia sets in and the Daily Prophet begins the character assassination campaign, Dumbledore's political power drops like a stone. Combine this with Dumbledore's tendency to keep his inner circle small, and he's having a very hard time against Voldemort's followers. Due to the wizarding world's attitude towards nonhumans and muggles, He'll probably be an in even worse position than canon, politically. However, as his political power lessens, he's probably going to be increasingly willing to involve Tenchi, Washuu, and the others simply because of the capabilities and firepower they bring to the field. Overall, Dumbledore's relationship with Harry and his adoptive family will never be too good, but it'll probably get better as the Voldemort situation worsens and at it becomes clearer that Dumbledore is more than a secret-hoarding control freak.
Dumbledore's other serious problem during the books is that outside Hogwarts, he really lacks any sort of official authority. If I remember correctly, he actively avoided taking positions of official authority because he was scared that he'd succumb to the temptations of power. Unfortunately, unofficial power requires people to listen to you- once's Fudge's paranoia sets in and the Daily Prophet begins the character assassination campaign, Dumbledore's political power drops like a stone. Combine this with Dumbledore's tendency to keep his inner circle small, and he's having a very hard time against Voldemort's followers. Due to the wizarding world's attitude towards nonhumans and muggles, He'll probably be an in even worse position than canon, politically. However, as his political power lessens, he's probably going to be increasingly willing to involve Tenchi, Washuu, and the others simply because of the capabilities and firepower they bring to the field. Overall, Dumbledore's relationship with Harry and his adoptive family will never be too good, but it'll probably get better as the Voldemort situation worsens and at it becomes clearer that Dumbledore is more than a secret-hoarding control freak.