Beta Multiplier Reference: What is going on with the current output?
It's certainly possible that exceeding the maximum \$V_{ds}\$ is causing the problem. I guess non minimum length transistors would help, but I don't know by how much (could be checked in simulation). I've done some simulations that suggest that \$V_{ds}\$ might be the cause, and should provide a way to check too.

I assume your simulations are based on the BSIM3V3 model (or some BSIM variant). If that's so, it has a parameter called PSCBE1, which has an effect on breakdown voltage. Obviously, the correct values and effect of this parameter will vary by process. For the transistor models I have, the effect of varying this parameter from 1e8 to 5e8 is as shown below.



I set up a similar Beta Multiplier to yours (not the same, a simpler one based on that in Baker's CMOS Circuit Design, Layout and Simulation). It's shown below. The 3 extra transistors to the right are just a start up circuit.



If I set PSCBE1 to 4.24e8, I get this response, which you can see is very similar to yours (it was a transient simulation, so I include a plot of VDD to show how it was being increased).



However, if I increase PSCBE1 to 7e8, I get this response.



You can see that the breakdown has moved up and off the graph. It's not a guarantee that \$V_{ds}\$ breakdown is the problem, but it certainly seems it is possible. One way to check might be to edit PSCBE1 in your own models and see if your problem goes away. Obviously, this means that your models then don't match your process, so you won't be able to to manufacture the device. In that case, you'll just have to accept that the voltage range is limited in your new process. Of course, if it has no effect, then this explanation is wrong and it must be something else!