Bluetooth 4 would require a hardware upgrade. There are two options you could consider, one would be to get a USB adapter the other more interesting option is to replace the existing internal Bluetooth card. I have actually done this myself and such an upgrade certainly does give you full 'Continuity' compatibility.
Mac Bluetooth Update
First see these articles. http://www.macworld.co.uk/how-to/mac-software/get-continuity-handoff-airdrop-on- old-mac-3582632/ and http://www.cultofmac.com/306681/add-continuity-mac-using-just-cheap-bluetooth-do ngle/
Bluetooth driver free download - Apple Bluetooth (Panther), Digidesign CoreAudio Driver, USB Audio ASIO driver, and many more programs. Get USB WebCam drivers for Mac OS X. Free to try User.
Bluetooth 4.0 Adapter For Mac
Second see this website. http://www.osxwifi.com/adapters/apple-broadcom-bcm94360cd-802-11-a-b-g-n-ac-blue tooth-4-0-with-adapter-for-macpro-2009-and-macpro-2010
Bluetooth Software 1.5 for Mac OS X: Information and Download This document contains the Bluetooth Updater 1.5 information and a link to the Bluetooth Updater 1.5 software. When you have a Bluetooth enabled Macintosh you can use Bluetooth technology to communicate wirelessly with devices such as mobile phones, printers, keyboards, PDAs and more. Mac OS X Support Mac Software Other Apple Hardware. No, the reason this bluetooth adapter is recommended is because it does not need any external drivers.
Bluetooth Adapter For Mac Os 10.12
The adapter above replaces both the original WiFi card and Bluetooth card with a current 802.11ac WiFi card and current Bluetooth card as would be fitted in the current iMac as an example. Since this is a genuine Apple part it is automatically recognised and supported by OS X. This is the approach I used in my Mac Pro.