I have an older version of Pro Tools that requires Mac OS v10.7+ but isn't capable of understanding x.10 is higher than x.7.

Therefore, the only thing left to do is buy a new copy (no) or change the Operating System's version number. Here's how:

Step 1 - Disable System Integrity Protection

A. Restart your Mac and boot it in recovery mode by holding down CMD + R.

B. Open up Terminal and type

 csrutil disable

C. Restart the computer.

Source

Step 2 - Modify your Mac's version number

A. First things first, let's create a backup:

sudo cp /System/Library/CoreServices/SystemVersion.plist ~/SystemVersion.plist.bak

B. Open up SystemVersion.plist using nano and set it to write mode.

nano -W /System/Library/CoreServices/SystemVersion.plist

C. Modify these two nodes:

<plist version="1.0">
<dict>
...
        <key>ProductUserVisibleVersion</key>
        <string>10.9</string>
        <key>ProductVersion</key>
        <string>10.9</string>
</dict>
</plist>

D. Save your work and install Pro Tools.

E. Once you get passed the initial installer, don't forget to undo your work by repeating steps 2.b and 2.c.