Yes, my Mac pro shipped with Apple keyboard, trackpad and mouse, but I've found them to be overpriced garbage (design and function) & don't use them at all now. Same for me with the poor bluetooth connectivity issues.
I now use a
Dell Premier Wireless keyboard (superior to Apple's in every way including ergonomics, feel, angle, key travel, switching etc) and a
Logitech Ergo wireless trackball, also far superior to the dreadful Apple mouse, eg, the Logictech can charge from USB while continuing to be used, switching, control IO etc.
Both the Dell & the Logictech are firstly connected to a Windows workstation via their wireless USB dongles. The Dell keyboard provides three way switching (1 x wireless & 2 x bluetooth); the Logictech mouse provides two way switching (1 x wireless & 1 x bluetooth).
The main point here in terms of the OP: unlike the Apple devices, both the Dell keyboard and the Logitech work flawlessly with the Mac Pro's bluetooth connection. This would seem to indicate that the problem lies with Apple's IO devices, not the computer itself.