What you did with the op-amp looks good.
The J-FET has problems.
Old circuit: D=1/2VI+, S=1/2VI+, G=oV
New circuit; D=0V, S=0V, G=0V. Problem is there is zero volts G to S.
I don't know what VI+ is. Pretend 10 volts.
1) If VI+=10 and VI-=-10 then: Use R1, R5, C4 to make a -5V supply and connect the ground point (R7, C5, D2) to the new -5V. This way the G to S voltage will be 5 volts again. (you moved the D and S down 5V so move the G down 5V) or 1/2 supply.
2) If VI+(old) is twice VI+(new). VI+old=10V and VI+(new)=5V, VI-=5V.
Connect the ground point (R7, C5, D2) to the new -5V. (-supply)
This way the G to S voltage will be 5 volts again. (you moved the D and S down 5V so move the G down 5V) or 1/2 supply.