The DS2 being constructed much faster than the DS1 really is not implausible at all. Of all the silliness in Star Wars, this is one of the least implausible things.
There’s many ways to easily explain this. It really depends on why the DS1 took so long. Firstly, the DS1 was developed secretly under a government that still nominally had some democratic organs and accountability to a pluralistic Senate. So planning, logistics, allocation of funds (a massive amount of funds!), would all need to be carefully orchestrated to appear harmless to the Senate, while also balancing the various competing interests inherent to any pluralistic governing body. But when constructing the DS2, the mask was already off. At that point, Palpatine was just like “Yeah, fuck it, I’m evil. Just deal with it.” So presumably there was no need to cleverly hide resource allocation to maintain a façade of accountability. Palpatine could allocate funds with dictatorial fiat.
Consider how in real life, planning and building the new World Trade Center tower in New York after 9/11 took over a decade, with around 8 years required for construction. Meanwhile, in the United Arab Emirates, a much larger tower, the Burj Khalifa, was constructed in only 5 years. This is because the construction companies involved in building the new WTC had to contend with the large bureaucracies and competing multi-agency interests in New York City, whereas the Burj Khalifa was constructed in a mostly monarchic (or oligarchic) state, with fewer competing interests.
Secondly, it’s also very plausible that part of the reason the DS1 took so long was due to the novel engineering and technical obstacles they encountered while building it. These obstacles needed to be overcome and solved for the first time, requiring extensive R & D. But once solutions were discovered, they knew exactly what to do the second time around. Perhaps they ran into technical difficulties with (I don’t know) heat diffusion caused by the superlaser. A lot of R & D would be required to solve this issue. But once an effective solution was discovered, they could easily replicate it a second time. This sort of thing is common in real-world engineering projects.
And thirdly, there was way more urgency to finish the project quickly the second time around, because after Yavin, the Empire was likely facing countless uprisings, badly straining the Imperial navy. The Empire played the “Tarkin doctrine” card but were then suddenly caught with no Death Star to back it up. They probably had to scramble to build a new one, while their military was stretched thin quelling endless uprisings.
So it’s really not particularly implausible to me that the DS2 could be constructed much faster than the first Death Star, because it was constructed under different political constraints with way more urgency, and they already solved all the engineering challenges required to create it the first time.
And finally, remember that the DS2 wasn’t actually ever completed. Presumably it still couldn’t move. And it was never demonstrated that the superlaser was yet capable of scaling up to the energy level required to blow up a planet.