Replacing a piece of nailed tongue and grooved engineered hardwood located in the middle of a large room
You are heading the right way with your first thought, although I rip out the middle 3/4" and chisel out the  ends where the saw will not get to. Pulling out the center will allow the saw to stay away from the nails or staples holding the piece down and will let the groove side slip out. Use the chisel at an angle to get the remaining stapled in part by splitting the flooring at the same angle the fasteners go in.

Yes do remove the bottom half of the groove side and glue it in place. Apply wood glue to the tongue and set construction adhesive at the tongue of the board in place so when the new piece goes in the part where the cut off groove is will grab the glue and keep it there.

You new piece should go right on in there. The original one did. and as long as the new one has acclimated to the new area it will go in, that should fit too.