According their site, the mouse works with 'Bluetooth Smart Ready device'.
We've put everything you need to get started with your Wireless Mouse M510 right here. If you still have questions browse the topics on the left. Check our Logitech Warranty here. Logitech Bluetooth Mouse M557 is the one mouse you can rely on to work with whichever platform you work on. Connect to any Bluetooth-enabled PC or Mac computer.
http://www.logitech.com/en-us/product/mx-anywhere2 ( Check System Requirements in TECHNICAL SPECIFICATIONS section )
Bluetooth Smart Ready on Mac means Bluetooth 4.0 hardware with Yosemite or later.
Unfortunately BT on your Macbook Pro (mid 2010) is BT2.1+EDR
( If you want to be sure, Logitech provides a procedure to check the PC/Mac is Bluetooth Smart Ready or not )
Logitech Bluetooth Mouse Setup
![Bluetooth Mouse For Mac Logitech Bluetooth Mouse For Mac Logitech](/uploads/1/1/9/8/119876204/890948701.jpg)
Best Bluetooth Mouse For Mac
I guess your only option is use of USB hub if you don't have available USB port,