There is always a challenge in moving from 'laboratory expression' to industrial expression. Most predominant reasons: oxygen limitation in your bioreactor (due to high biomass conc. of P. pastoris--you would need to supplement air with pure oxygen to overcome this), plasmid shedding (due to high generations in bioreactor), inadequate mixing and others.

In terms of your BS medium, I used 1/2 of the concentrations specified in the Invitrogen protocol to get higher expression in bioreactor. It does happen that that medium is highly over-specified (my paper on this is coming out soon in Bioproc. Biosyst. Eng. J).

So you can play around with your medium, but also check the oxygen supply.

Yes, the main challenge in scaling up is oxygen supply when you have to keep the DO>20%, first you have to compare your cell concentration in the lab-scale with the one in scale-up fermenter at DO>20%, most likely in the scale-up fermenter your cell concentration is lower and because the overproduction is typically proportional to the cell concentration you are having lower level of your target product.

To increase oxygen supply in the scale-up fermenter ( increasing mass transfer of oxygen) you can use the maximum impeller speed, the maximum air flow (>1 vvm) and the maximum pressure that your fermenter can work (at least 1 bar that is sterilization pressure), and the lower temperature will also increase solubility of the oxygen as a gas.

You need to optimize your medium by Surface Analises Response methodology were you can include the effects of DO together with differentes sources and levels of carbon, salts and  nitrogen ingredients using as a response the invertase enzyme production.