I read the service I linked you to. It had a German Link name but it is in English. In the service guide, they show the bearings being pulled using a slide puller. They have fingers that go behind the bearing and the weight slides along a shaft and and pulls them out. The image in the page I linked you to does the same thing just using a screw instead of a slide weight. Both do the same thing. Whatever you can get will work.
It is possible that your wheels have a "detent" or groove made into the wheel that would prevent the bearing from being pushed thru the wheel using a drift ... I don't know. What this means if you tried to push them, you are only making yourself more work. They say to pull them, I guess we pull them. It does say the wheel has two bearings and a spacer shaft between them. It is possible the machine shop could pull the old bearings and press the new ones in for you but that costs money... but usually not much and less for cute girls. Total up the cost of the puller (cost or rental), transportation to the place to get the puller, time back to the wheel, pull time, press time, time to take the brake disk to the machine shop, time to ask about the disk...
The damage to the brake disk may look worse than it is.
Last time I went to the machine shop for a small project, i was charged the shop minimum, $25.00. It might be worth a phone call. Be honest but don't waste their time either. Respect their time and answer the question quickly and honestly. if you don't know, say so and don't give them a BS answer.