After doing some research, I found no ContextActionService bindings to backspace.

I also found a forum post saying the Backspace to drop tool is implemented in the system and they couldn't move it into a PlayerScript due to the potential of breaking games.

Solution

It seems that parenting it to the workspace is your only choice here. I tested it in-game and parenting it to workspace acts the same as pressing Backspace (that is while it's equipped), so it basically has the same effects.

You could also make your own backpack system and make a function that unequips it. Maybe create your own toolbar (disabling it through SetCoreGuiEnabled) or use Tool.CanBeDropped to disable it from being dropped and add your own functionality to it. If you want it to drop in front of the player, you'd use CFrame.LookVector to base it off the player's orientation and whatnot.

If you want to unequip a tool that's in a robloxian's hands, then you can use:

plr.Character:FindFirstChildOfClass("Tool").Parent = game.Workspace

or if you want to unequip all the tools of a player's backpack:

for _,tool in pairs(plr.Backpack:GetChildren()) do
    if tool.ClassName == "Tool" then
        tool.Parent = game.Workspace
    end
end