The issue is 100% custom content (including vehicles). In simple terms, LSPDFR is designed to use the headroom offered by vanilla. LSPDFR works absolutely fine with said vanilla content. Custom content lowers, and/or removes this headroom. The issue could be replicated in 0.3, it was just not nearly as common, as 0.3 didn't require as much headroom as 0.4 does. Such is trade-off of new features.
The answer you're looking for, is to open the custom vehicles, and reduce the texture sizes. Many people suggest targeting the sizes of the skins (From 4k to 2k is a good start), however any texture file included with the vehicle is a viable target, as interior textures on them tend to be excessive as well.
The actual amount of reducing you need to do, will depend on the specific cars you have installed (Some are higher poly then others, and some have more unique texture surfaces then others), and the amount of cars you have (Having 6 different vehicles all spawn in the same area is more taxing then 5 variants, etc.). Any other custom content that goes above and beyond the vanilla content (Guns, general textures, etc.) will all contribute to the issue, cars just tend to be the most common dominater.
If you cannot find a satisfying combination of reduced textures that gets rid of the issue, it's time to start trimming the custom vehicles you have installed just for varieties sake. At some point, you will hit a happy medium where texture loss either doesn't occur, or takes so long to occur that you don't run into the issue.
I've also seen someone mention turning down the texture settings ingame to compensate, and that solving the issue for them.
You need to stop thinking in terms of what worked fine in 0.3.
0.4 has a new ambient spawn system that likely eats up more memory, I wouldn't be surprised if EUP peds are more taxing then regular peds, and 0.4 allows a person to easily have more vehicles beyond the vanilla slots all spawn in the same area. At some point, you just start hitting the limitations of the game. (For reference, I ran into the texture pop issue in 0.3, it is not an issue solely caused by 0.4, it's a memory issue on the games side)