This isn't answering your question directly, but on the CD case for the original Starcraft (should be the first disc) there should be a 13 digit cd key.
If you try downloading the Starcraft installer from Blizzard, you wont require the discs to play the game. You will be able to launch them straight from your desktop. However, that game requires a 26 digit key. Blizzard allows you to register you 13 digit cd key to an account to receive a 26 digit key which you can use to use the downloaded installer.
You can log in or register at https://us.battle.net/login/en/. Once you log in select the "Games & Codes" menu and from there select "Add a Game Key." Enter your 13 digit code and you will then receive your license key which allows you to download the game.
On a side note, when I bought my Battle Chest (12 years ago), the first disc was StarCraft and the second disc was the Brood War Expansion. Starcraft should install independent of Brood War while Brood War depends on Starcraft, much in the way that Heart of the Swarm and Legacy of the Void expansions for Starcraft II require that you own Starcraft II: Wings of Liberty. It may be the case that the newer Starcraft BattleChest comes with a different set up now, but if it was like how I had it, you shouldn't need to switch discs.