Electronic – LMC6062 Jumps too Rail when 1V below

I think you completely mis-understand what a 4-20 mA loop is.

In your circuit, even if you could get the current to 20 mA through the R1 resistor you could not drive an external loop Rx with it.
Most 4-20 mA loops use an external power supply for loop power and may have significant voltage drop in their Rx circuits and connecting wiring.

You should study devices such as this which produce an accurate loop current, though it's supply voltage for the op-amp is much higher than 5V:

Your circuit might work better like this, though the input voltage is 0.4 - 2.0 V for 4-20 mA:

^{simulate this circuit – Schematic created using CircuitLab}

The FET has a very low VGS(th) and the output loop could only withstand a 12 V loop supply.

The oscillation is a result of instability in the layout, perhaps inductive power or ground leads without decoupling 🧢’s on power to gnd .

Unity gain gives the widest BW (from fixed GBW) but also the lowest phase margin which degrades with any capacitive load, unless compensated with reduced BW.

There is also a slope on the pulses, which suggests another reactive effect in your layout not shown on your schematic but could be uncalibrated 10:1 probe error.

So correct that with the probe trimmer and scope test pulse , add ceramic caps. to both supply rails and use a very short probe ground lead to get textbook waveforms from DAC then OpAmp.

Even Alkaline batteries make random noise which you can hear with 1.5V on a speaker, spurious effects depend on the phase margin which can improve with gain or in your case be asymmetric while improving.

Define your capacitive load! Is there a long cable output and measure your 9V supplies with a divider.