The friend, who has previously assisted me in looking for the rodent and bubble game I remember from my early childhood, pointed out that it's possible that I might not be able to find the game, because many games from this time period have not been well preserved.

They linked the paper Survey of the Video Game Reissue Market in the United States:

The results are stark: Only 13 percent of classic video games published in the United States are currently in release (n = 1500, ±2.5%, 95% CI). These low numbers are consistent across platform ecosystems and time periods. Troublingly, the reissue rate drops below 3 percent for games released prior to 1985—the foundational era of video games—indicating that the interests of the marketplace may not align with the needs of video game researchers. Our experiences gathering data for this study suggest that these problems will intensify over time due to a low diversity of reissue sources and the long-term volatility of digital game storefronts.

Specifically, for the period of 1990 to 1999, roughly the game as the time period at the game in question I'm looking for, the study found the following data:

In release. For the purposes of this study, a game is considered to be “in release” if the game, or a version of the game derived from its original release—including emulated, modified, or re-implemented versions of the original game—is reasonably, readily, and legally available from the game's rightsholder, either in physical or digital format, for a currently produced or supported game platform. This does not include remakes of the game or versions of the game that are substantially different from the original.

So something like 80% of the games from this time period are not reasonably, readily, and legally available.