Whew, getting up at 330am would not make me happy either! LOL.
I got Jackson when he was 9 weeks old. He had a lottt of energy when he was younger. Well, he's 9 years old and STILL does (but I love that about him- as I wanted a hiking/walking partner, etc) but he has much more of an "off switch" now.
I taught Jackson early on I wanted my sleep though. To be honest, at 7 months old, he should not still be getting you up in the middle of the night. Is he crated, or what are his sleeping arrangements?
Also, besides physical exercise... something to mentally wear your pup out... check out some puzzle type toys. They have many toys that make the dog have to WORK for their food and it lets them use their brain and get self-rewarded and can keep them busy for a bit.
A regular Kong toy is always great. You can stuff it with treats and then use peanut butter to fill it and freeze it for a long lasting treat.
Kong Wobbler is a fun food dispensing toy that the dog has to learn to knock over to get the food out of the little hole.
Nina Ottoman toys are good.
Just some suggestions for you.
I definitely think training will help strengthen a bond.
Also- I can understand being overwhelmed by a Yorkie coming from a Shihtzu. Every Shihtzu I've been around (I'm a dog walker) has been SO laid back. Like just the most chill little things ever. They'll go for a little stroll and that satisfies them. Most Yorkies are like little energizer bunnies especially when they are younger.