AAA battery to power a usb device?
The product at your link will not work for your problem.


The principal issue is that it is a USB device, not a USB host.  Two USB devices (this and your mouse) can't talk, unless one of them is able to switch modes and be a HOST.
Further your host has to have software that knows the USB mouse protocol
That is a BTLE device not classic Bluetooth; it may be that there's a way to do HID mouse over BTLE, but it's at minimum atypical
The product description is written from a user rather than technical perspective; BTLE does not have "master" and "slave" but rather "Central" and "Peripheral"
And yes, you still have the power problem.


If you really want to build this you could try using an MCU that can be a USB host - often there's a mouse hosting example for those.  Then you'd need a classic BLE module with an HID firmware, and hopefully one that connects over UART or SPI.

Or you could use a more capable MCU or small computer and see if you can get both the mouse and bluetooth connected as USB peripherals (though I don't know if common USB bluetooth dongles can be HID devices, and you either need two USB host engines or to add a hub)

It might be better at that point to go to something running an operating system that already has the relevant drivers - an embedded Linux board, or even the PC you are sitting at as a proof of concept. 

You could also consider getting a PS/2 mouse; those have a much simpler synchronous interface that you can bit-bang on any pair of GPIOs perhaps with level shifting circuitry.

Or you could buy a bluetooth mouse...

As for the power issue; a boost converter if your peripheral is low current, but don't expect long lifetime without some very careful engineering of power usage and sleep modes.