The picture of an old CRYSTAL bird is probably the closest thing to the OP in terms of difficulty. The satellite is big and in LEO. The difference between what Ralf did then is that CRYSTAL satellites move less often, so once someone finds them you have more time to wait on good weather. With the X-37, Ralf missed his first shot and by the time he was ready for another try it had already moved, and they had to find it again. That is super tough.
If you have no idea where something went, then the sky is very big and your telescope has a very small field of view, especially if you need to discriminate between different satellites that you see. The fact is, there aren't many techniques for finding secret stuff in orbit. You have two methods (but choose a wavelength)- passive and active- and maybe hacking into a space agency and just stealing the information from a computer. Passive means staring at a lattitude long enough until your satellite passes by and hope you spot it (if it doesn't you either missed it or it has a lower inclination than here you were looking). A smart person might get a computer to help them with the watching, but the task is the same. The problem is that the X-37 is real easy to miss. Even if you see something, it might not be the particular satellite you're hoping for, so you have to filter out all the unwanted things you don't want (usually you can match unwanted things up to known orbits). And even when you see the X-37, you've got to jump on it and track it so you can figure out the new orbit. Active methods are pretty much just radar, and that's out of reach for every amateur astronomer. They might be able to scrounge up some giant apertures but they'll never have the tx power. It's pretty incredible that people are able to do this.