+-5VDC to 4-20mA Converter
I don't recommend you to use the design in the picture. OAs ( Operational Amplifier) are good for Voltage gain and High impedance output, but not to "play" with current, the current output is sensible with the components tolerance. Try to simulate your circuits with variation of 5% in your devices (resistors &amp; VCC) and see the output. When you will buy the components they will have deviations about 1% to 5%, depend on the quality of the materials you will buy.

Transistors are the devices to play with current gains, but impedance could be the problem. 

See also the tolerance for the Valve mid-position for current, it is 1.3%:

Voltage type [V DC] 5 (±0.1)
Current type [mA] 12 (±0.16)


Your are right the 12.267 mA are not correct ( max. acceptable would be 12.16 mA), take this into consideration because it means: the tolerance current output of your circuit must be max. 1.3%!!!.

It could means also that maybe temperature compensation of your output must be implemented depend on the range of temperature where the device will operate and ICs you will use.

I have seen the Page 6 Datasheet of the valve: 

Setpoint value
Voltage type [V DC] 0 … 10
Current type [mA] 4 … 20


Why don't you try to control the valve with "Voltage type"? I think it is easier to transform your +-5V to a 0-10 Volt linear signal with mid-position on 5V. And you have a little more tolerance (2%) requirement for your output. It is easy, you can use an OA adder with a two inputs: one a +5V offset and the second your +-5V signal (you must check the tolerance).