Budget Driver Board than can do 5 Axis Kinematics ?

Kinematics is not "done" on any board, it is done in LinuxCNC, so any 5 or more axis board will do.
Download the official ISO linuxcnc.org/downloads/
write it to a USB, boot from that USB and play with the 5 axis configs that are included, no need to install it onto a hard drive.
CAM software for 5 axis is very expensive, there are some free versions that might be usable but i never used them.
Build the machine, later deal with software and cam.

FreeCAD < not sure how many axis
BlenderCAM < not sure how many axis
PyCAM < 3 axis with multiple passes

Ruddy wrote:

Are you sure fusion 360 3+2 is free ?
 

Recent email from Autodesk:

Effective March 29, 2022, updates were made to the capabilities provided in the Fusion 360 Manufacturing workspace.

We've simplified how we organize and offer these capabilities, while including more as part of your base Fusion 360 subscription. 

What's happening?
All Fusion 360 subscribers will gain access to extra 3+1 and 3+2 capabilities (4- & 5-axis positional), at no additional cost. These features were previously only available in the Machining Extension and include interactive tool orientation controls such as align to view, align to surface normal, and interactive tilt and turn.

At the same time, more advanced and specialized 5-axis simultaneous features will only be available with a subscription to the Machining Extension. These include swarf machining, multi-axis contour, multi-axis tilting (for 3D contour), and multi-axis flow functionality.