Sorry Rust, but the Soviet Union leadership did believe the Star Wars claims.
Why? Because the US didn't just come out and make these claims. In fact they tended to publicly release information that indicated that it was failing while apparently trying to conceal any signs of success.
To many people in the world (including experts), it looked like the US was concealing successful tests and showing only the failures. They also gave secret briefings to thousands of soldiers showing working prototypes of laser equipment along with other information that was actually true, knowing that the information was going to leak out (I was personally at one of these briefings, and it was very convincing at the time.).
This was not a haphazzard effort on the part of the US, it was a well planned and well executed plan (This actually suprises me.).
There were always detractors, but there were people who called themselves experts who didn't believe in the NSA or Stealth technology either. Things were deliberately confused in order to create doubt as what was the truth.
You need to realize that the Soviet Union couldn't afford to take the chance that the technology actually worked, because if it did their ICBMs would be useless, while the United States' ICBMs would still be effective (Quite Unacceptable) . So they had to try and find away around or duplicate a technology that didn't actually exist. They were already on the edge of financial disaster, and this pushed them over.