Upgrade bluetooth on Mac Pro 3,1 in order to support Surface Keyboard

I bought a Microsoft Surface Ergonomic Keyboard for use with my Mac Pro 3,1 (early 2008 quad core). It requires Bluetooth 4.0, which I thought I had -- under System Information, it tells me I'm running Apple Bluetooth Software Version 4.4.6f2. But it does not even show up when the bluetooth preferences panel is looking for devices to sync with (yes, the keyboard is in sync mode).

I'm running El Capitan 10.11.6. I'm aware of the hacks for running Sierra and High Sierra, but for a variety of reasons, I'm not willing to try installing them (again). The hack is not compatible with a couple of the upgrades I've got on this system, including OWC's 64GB memory modules and a 2TB 6G SSD on an internal PCI-e controller (macOS installers don't like to install on external drives, which they think my SSD is because it's on the PCIe controller).

Now, reviewing the original specs for the Mac Pro, I see I may have been mistaking hardware and software versions. The original specs note the 3,1 has "Bluetooth 2.0+EDR", which I assume refers to the bluetooth radio/on-board hardware.

So question is... if we assume I keep running only El Capitan, how can I upgrade? Is the internal Bluetooth receiver replaceable? Is it a part of the wifi card? Soldered to the logic board? I do not use WiFi on the machine; the machine runs CAT6 cables. And I'm not opposed to soldering things if needed. Just trying to figure out what my options are.