Currently, I have a 400W PSMH Westinghouse lamp made in China, and I also have a GE M400 in 400W HPS, which are obviously two different lamp technologies. Recently, I was wondering what would happen if I ran the 400W PSMH lamp on the 400W HPS ballast. The arc voltage drop is higher than that of a 400W HPS lamp, so how would this affect the ballast? Would it overload the ballast?

Is your 400W ballast of HX or CWA topology?

The "turning purple" is just one particular failure mode on which just the particular LED design used in the streetlights was prone to. I don't think that style is used anymore, not just because of the tendency to turn purple, but also because the LEDs are extremely vulnerable to damage during assembly.
For domestic LEDs a broken bonwire failure (LED chaotically flickering or cycling, then dying, often with a noticeable internal burn mark) is the most more common failure mode these days.
It seems most problems become solved, except the bonwires. It seems there is no principal solution except some minor optimizations here and there.

It is matter of selecting particular unit power size for the given task. Along with the maximum power, to ensure it would be able to keep up with the mose demanding condition.
Obviously you won't fit exactly, but you should aim close, selecting a model/rating variant that is the closest.with its performance.
Practically it means how the manufacturer matches the exact performance with the normally used power rating category and the climate of the target market area (each exact AC designs are tailored for the given territory separately, combination of the temperatures, winds, humidity and so on, so a unit sold in one place may perform very bad somewhere else just because it is set for different climate).

I have seen the opposite thing happen once

The diffuser material on the LEDs ("1W"/"3W" style gull wing LEDs) degraded from the intense blue content in the light, discoloring to a burnt brown color and making the light color shift to some orange-ish color as well as drop in brightness

This happened in lamps that been some 12+ years in service though, most lamps just dont last anywhere this long

The level at which the unit must run (including above minimum) would vary with each individual installation. It is not something the unit designer can predict with high precision

I second Multisubject's answer

There are PSMH lamps for the European market rated for this exact use (400W HPS ballast), but there the ballast is just a choke so there is no primary winding to overload

500W is a significant overload, so unless the ballast is running in a significantly reduced ambient temperature (able to compensate for ~20W ish additional loss), this will probably lead to overheating of the ballast

A ballast of this size has high thermal capacity, and may take well over an hour to stabilize at steady state temperature for measurement

Lowe's when they were still around in Canada started discontinuing HID lamps in 2012 or so by cutting down on the range of lamps offered. By 2014, Lowe's Canada pretty much stopped selling HID lamps. I got a few cheap HID lamps at the time when they were clearing out the stock, mostly newer Sylvania lamps.

In late 2023, Lowe's spun off the Canadian division along with the Canadian RONA stores they owned. The Lowe's stores were rebranded under the RONA banner by early 2024. 

Going by reference ballast characteristics (not exactly realistic, but good enough), the 400W HPS needs 38.3 ohms at 220V while the 400W MH needs 45 ohms at 220V. That is a 16% difference, so not insignificant.

According to my Desmos calculator which has a history of being semi-correct, running a 400W MH lamp on 38.3 ohms at 220V will run it at almost 4 amps, which is a 23% lamp overdrive, assuming that the lamp voltage doesn't significantly decrease from the normal 135V (not exactly realistic, but good enough). Very significant lamp overdrive, but probably fine for short periods or with a lamp you don't care about.

This would mean that the ballast meant for ballasting 400W lamps is now ballasting a lamp that is operating at more than 400W (probably around 500W if I were to take a wild guess). The secondary coil would probably be fine since it would be carrying less current than normal, but the primary will probably carry more current than normal due to the increased secondary wattage. If you choose to do this, just monitor the temperature and make sure it doesn't get too hot I guess. Long-term it will certainly reduce ballast lifespan.

Currently, I have a 400W PSMH Westinghouse lamp made in China, and I also have a GE M400 in 400W HPS, which are obviously two different lamp technologies. Recently, I was wondering what would happen if I ran the 400W PSMH lamp on the 400W HPS ballast. The arc voltage drop is higher than that of a 400W HPS lamp, so how would this affect the ballast? Would it overload the ballast?

That's interestin. What function does the carbon dioxide has in these lamps?

Good question!  I am also not entirely certain, but suspect it helps to waste some energy in the form of heat at low current operation, so as to help keep the electrodes warm enough to prevent sputtering.