When the runway required as calculated under reported conditions approaches the runway available, it just doesn't take much variance in actual conditions to get this sort of outcome. As well, any variance in technique or execution from the assumptions inherent in the performance calculations invalidates the predicted performance to some degree.
With the above in mind, heavy rain associated with a cold front is often accompanied by changing wind direction and velocity during approach. Some amount of shearing in both horizontal and vertical planes is to be expected. This often makes staying on speed and profile during approach somewhat more challenging. The FDR data will shed some light on how much a factor that may have been. I'm just saying that arriving at the TDZ on speed and profile is more challenging in that sort of weather.
I'm also curious as to the location on the airport where the reported wind measurement was taken. They have sensors all over the airport for the purpose of LLWS alerting. While there is no mention of any wind shear advisory having been issued, that does not necessarily mean that the wind at the end of rwy 08 was the same as the wind stated in the report.
It's been many years now since I was based at BUR, but I certainly recall that the winds were always more variable and gusty when out of the West through North. "Normal" winds always seemed to 180/10! KBUR's location with respect to the nearby hills in all quadrants probably has allot to do with the "swirl effect" that so commonly rears it's ugly head when the wind kicks up.
In the three years I flew out of BUR, I only landed during rain storms a couple of times.The rain was only moderate. And that was in a jet that required only about 3/4 of runway distance that a modern 737 at typical loads does. Maybe we were just being overly conservative in some people's view, but we were encouraged by the management to use an 80% (or 125% when looked at the other way) standard and that was before the 15% safety factor was even "a thing". Every pilot has used more runway than they planned to. It's just that most of the time, you don't land if it looks like you're gonna need all of it. So maybe something went way different than planned?