Never try to update to iTunes 12.7 as it is a complete failure. With it, you will lose the ability to locally manage apps and tones. Backup and restore may work as expected but all apps are not available through iTunes, only built in App Store.
iPhone 6 is surely supported by iTunes 12.6. However, given your current situation, I strongly recommend you to not update your Mavericks until you have the new drive bought, tested and you migrate all of your data. It is even riskier to break daily workflow than worrying about security risk or something similar. There are cases where musicians cant update past Mavericks for some similar reasons (critical software only supports Mavericks). Wait until you can migrate your data.