I have a similar configuration to you except for a BT mouse and SSD for Mojave. My Mac Pro 3.1 does not have the sleep issue. It wakes up with either the BT mouse or apple usb keyboard. However, Mojave does not always seem to play nice with the Mac Pro's usb; 14.0 seemed fine but not 14.1 or 14.2 in my case.There have been several comments about this on the thread. In my case, the usb keyboard did not work during the start up so I was unable to get the boot screen etc. Once Mojave was up and running everything was ok. I had connected the keyboard via my apple LED cinema display and the usb hub did not see to be powered up during the start. Connecting the keyboard direct to the Mac Pro solved the problem. So if you are using a usb hub try connecting the keyboard directly to the Mac.
It may also be worth looking at your Energy Saver in preferences. If "Put Hard disks to sleep when possible" is ticked then untick it to see if that helps.
When I have run into this recently, it was solved by going to the Patch Updater and forcing the reinstallation of all of the patches (which I assume cover the legacy usb patches under another name now). In my case, the manifestation of the problem, on a MacPro 3,1 with Cinema Display usb hub handling the wired keyboard and mouse, was losing access to those input devices after ejecting usb storage devices.