I'm assuming you want the center portion that is 0.5" wide to be on a different center than the ends. Couple of ways to do it come to mind.
1) Put the work in your 4-jaw and use an indicator to dial in jaws 1 & 3 so they offset each other by the difference in centers. Your 4 7/8" dimension is off, I'll say it's 3/8" for the example. Zero out on jaws 2 & 4, but adjust 1/3 so they read off from each other by 0.125". Center drill the end and turn down the middle section.
2) You could put the 1/2" round in a V-block and use a surface gauge or better yet a height gauge to scribe a line on both ends in the middle of the round. Then turn the round 90 deg. in the V-block and scribe a cross line at 3/16" above or below the top or bottom edge of the part on both ends. Then center punch both ends. Use your 4-jaw and a wiggler with a braille point in the tail stock. Put an indicator on the edge of the wiggler and adjust the 4-jaw so the needle doesn't move as the chuck is rotated. Remove the wiggler and center drill. Repeat on the opposite side and mount the part on the lathe between centers and use a dog to drive it.
Guess an alternative to 2) would be to mark just one end with the offset center/ center punched hole. Use the wiggler in that hole and zero out the 4-jaw with the indicator on the wiggler. Your part is only 1.5" long, so not too much sticking out of the chuck.
Hope this makes sense.
Bruce