The issue is reproduced in Original Firmware.
Slim chance of Rockbox being able to do better then, I'm afraid.

(Is Rockbox really that "unstable" on the c200 v2 though?)
AFAICS, with standard manufacturer-suggested settings we have about 70 kHz of IF bandwidth in this low-IF design (there
is a register to set this), and I've found no hints of any deviation compression (Ã la TDA7000 family) being applied in this chip.
70 kHz is not a lot. It would actually work reasonably well here in Germany where standard deviation is 40 kHz, but not too well with any large deviations.
The only workaround I can offer is turning down the volume at or before the FM transmitter so that deviation decreases.
Looking at other chips, especially Philips/NXP TEA57xx, it seems the LV24xxx employs an unusually low IF of 110 kHz typical. The TEA5767 is a 225 kHz design with a typical 94 kHz bandwidth, and a TEA5777 gets 110 kHz of bandwidth out of 150 or 133.333 kHz.
Even if we go up to 0.8 for BW/IF here (manufacturer recommended value is .65 for the LV2400x), that would still get us about 90 kHz only, with uncertain consequences in term of selectivity and image rejection.
If you have a reasonably well-populated local FM band and feel up to the task of setting up a build environment for Rockbox (took me about a weekend with some general Linux / *IX knowledge), you could play with the standard value of the IF_BW register in lv24020lp.c and see/hear what happens. One really needs to have a device on hand to test this.