Indexed rear derailer shifts two cogs for each step I'm trying to troubleshoot poor shifting issues on a mountain bike that appears to have been built out of the spare parts box…

The current problem is "instability" on the rear derailer – the chain bounces around and is very hard to get to settle into gear. The original setup was a SunRace M40 derailer paired with SunRace twist grip shifters and a Shimano 8-speed cassette. The chain measures closer to 7-speed than 8-speed (0.285" a bit over 7.2 mm).

On the SunRace website the M40 derailer is described as being 7-speed. So I figured that I might have found the problem (7-speed indexing steps paired with an 8-speed cassette), so I tried fitting an NOS SRAM 5.0 8-speed twist grip shifter (which, according to the instructions that came with them, are Shimano compatible). However, after reading @Chris in AK's comments below, and failing to find any definitive marks on the shifter, I actually measured the cable travel using the information I found in this article on drive train compatibility and it looks like the SRAM grip shifters are ESP (1:1) as the cable movement is slightly more than 4 mm / click.

The good news is that the shifting seems to be more stable, the bad news is that it reliably shifts two cogs for each clink (e.g., I move the shifter from the 8 position to the 7 position and the chain moves to the #6 cog).

Any ideas about what could be going on?
I would check the SRAM grip shift that you purchased.  SRAM makes grift shifters for both their own derailleurs and Shimano's.  The actuation ratio on those shifters is different.  SRAM uses a 1:1 ratio for their high end stuff.  Most of Shimano's is closer to 2:1.  The SunRace website you linked to lists the M40 as 2:1, meaning you'll need a SRAM grip shifter with 2:1 actuation.