Lathyrus latifolius -- California

The species has naturalized where it is not native in the eastern Mediterranean region (Cyprus), Northern and central Europe, Australia, and North America.

Yes. The species has naturalized in 43 counties in the region of interest, California, USA. (Calflora)
As well, the species is naturalized in Oregon and Washington, which have a similar climate.

The plant is listed as being invasive in many areas of the US, primarily referencing roadsides and rights-of-way (Oregon Dept of Ag).
Considered to be invasive in some areas in the US (Missouri Botanical Garden)
The plant is an environmental weed in Victoria, Australia and on Lord Howe Island, it is listed as an invasive plant that is targeted for eradication. (Weeds of Australia)
This was noted as late-season monoculture in the San Francisco area. (R. Kesel, Pers. Comm.)

Within the region of interest, this was noted as becoming a late-season monoculture in urban areas around San Francisco, and it has been managed in another disturbed preserve in San Francisco. It was also noted that it demonstrates an extremely extensive rhizome network. The plant occurs in the front country of Mt. Tamalpais, Dipsea area, and several highway corridors in Sonoma County (R. Kesel, Pers. Comm.)
Within a similar climate, the plant is listed as being invasive in Oregon, primarily referencing roadsides and rights-of-way (Oregon Dept of Ag).
Although it is noted that it is mainly a weed of disturbed sites and roadsides where it occurs in Australia, they do mention that it has invaded natural areas as well. The plant is listed as being invasive in Victoria, Australia, some of which matches the climate in California. (Weeds of Australia, Queensland)

Lathyrus odoratus is noted to be invasive in New Zealand and possibly in the Dominican Republic (CABI). This species has naturalized in California but no information was available about it being invasive.
No other Lathyrus are on the noxious weed list in the region of interest.

The plant is found throughout vast areas of the United States, much of which does not match the region of concern. The areas where it is found in Central, Eastern and Northern Europe and Scandinavia do not match the region of concern. Of the areas reported in Australia and New Zealand, about half of the areas match the climate in the region of concern. (GBIF, Invasive Plants Atlas)

In Oregon, USA, the plant is known to overtop and smother native vegetation.

No effects noted having to do with fire. The plant is a low-growing tender perennial, so unlikely to contribute to fire risk substantially.

Can be toxic to livestock. (DiTomaso et al)
USDA GRIN lists it as a vertebrate poison for mammals.

The plant grows tender foliage and flowers to 4-8" tall, so this is unlikely.

Rhizomatous growth is a well-known method of reproduction for this species.

Since this is a vining species, it is likely that animals moving through the underbrush could easily detach pieces of the vine and deposit them elsewhere, however, this was not noted in any reference.

The plant is said to easily self-seed.

The primary means of reproduction appears to be vegetative. It was noted that seeds were rarely set in greenhouse settings, and seed set in the field is very dependent on pollinators and the abundance of nectar robbers (Godt & Hamrick 1992).
The number of ramets in the field far exceeded the number of seedlings. (Hossaert-McKey and Jarry 1992)
A study of ovule position in natural populations in Europe showed some variation, but note an average of 10-12 flowers per inflorescence, maturing 0.5-1.5 pods each, and an average of 5-6 viable seeds per pod. This source did not mention the number of infl. per individual, but the numbers would likely be less than 1000 seeds per individual. (Hossaert & Valero 1988)

There was no mention of specific requirements for germination. Germination rates were listed as high under various conditions listed in Kew Seed Information Database.

The species didn't flower the first year under controlled conditions in a study that also showed high mortality, so this is probably not typical (Hossaert-McKey & Jarry 1992).
The plant is listed in garden blogs/web sales as flowering during the first year, and also unlikely to flower in the first year. None indicated that it would take longer than 3 years for the plant to flower.

Calflora lists the bloom period as May-September (5 months). Other garden sites list the species as blooming "all summer."
"The first flowers of both species appear in late June and early July, and the last pods mature in October. Each of the pink flowers ...stays open during about one week. Fertilized fruits require about one month for maturation." (Hossaert & Valero 1988)

Being fairly toxic to wildlife and with the evolution of a violent dehiscion and vegetative reproduction as dispersal, it's unlikely.

The large, heavy seeds are noted to be primarily gravity dispersed. (Godt and Hamrick 1991)

This is noted to be a roadside weed in Oregon, however, there is no specific mechanism of attachment to vehicles, and it is not noted to be an ag contaminant. Dehiscing violently and gravity seem to be the primary way the seeds disperse.