Would it be possible to reduce a 3 foot length by half the height of one floor joist if it
Was sistered with a 3/8" thick by 6 foot steel plate bolted through the joist at the two
Unaltered ends? We're trying to find a few more inches of head room in our basement
For use of a treadmill. Running between two joist is going to be tricky!

I'd love to tell you it will be fine but only a strutural engineer can truly say that. Now if I were jury rigging something like this I'd give it a try but with more overhang than 1 1/2' on each side of the 3' reduced space. You'd think a steel plate is great because of its strength but each hole you drill in the joist weakens it and makes a new likely spot to start a split. I'd go with two 1/2" plywood planks 8' long glued and screwed on each side rather than a simple sistering. If the floor has deflection after you do the install then you can also add cross bracing to the joists on each side.

A long time ago, I could do these calculations. There are bolt selection and bolt patterns that have to be considered.