Thanks for this. This is an interesting angle that I hadn't really thought about before...
Reference on kava inducing CYP1A1:
https://academic.oup.com/toxsci/article/124/2/388/1685738
Benzo[a]pyrene is found in tobacco smoke, which by itself is bad for you. Is it possible that drinking kava could cause tobacco smoke to be more carcinogenic? I guess it's possible, since by inducing CYP1A1, more of the carcinogenic metabolite might be produced, although, on the other hand, the metabolic pathway that produces it has other steps in addition to oxidation by CYP1A1, which I don't think kava would enhance. So it would depend on whether the CYP1A1 step is rate limiting, which I don't know. Ref:
https://en.wikipedia.org/wiki/(+)-Benzo(a)pyrene-7,8-dihydrodiol-9,10-epoxide
So, tobacco smoke + kava = unknown, but theoretically might be worse than tobacco smoke alone.
Also, as you imply, benzo[a]pyrene is found in other things we consume, such as cooked meat, especially meat that is overcooked or charred. Should we be concerned about burgers + kava?
The wikipedia article on benzo[a]pyrene (BaP) say this about carcinogenicity:
"A 1996 study provided molecular evidence linking components in tobacco smoke to lung cancer. BaP was shown to cause genetic damage in lung cells that was identical to the damage observed in the DNA of most malignant lung tumours.[14]
Regular consumption of cooked meats has been epidemiologically associated with increased levels of colon cancer[15] (although this in itself does not prove carcinogenicity),[16] A 2005 NCI study found an increased risk of colorectal adenomas was associated with BaP intake, and more strongly with BaP intake from all foods.[17] However, the foods themselves are not necessarily carcinogenic, even if they contain trace amounts of carcinogens, because the gastrointestinal tract protects itself against carcinomas by shedding its outer layer continuously.[citation needed] Furthermore, detoxification enzymes, such as cytochromes P450 have increased activities in the gut for protection from food-borne toxins. Thus, in most cases, small amounts of BaP are metabolized prior to being passed into the blood.[citation needed] The lungs are not protected in either of these manners.[citation needed]
The detoxification enzymes cytochrome P450 1A1 (CYP1A1) and cytochrome P450 1B1 (CYP1B1) are both protective and necessary for benzo[a]pyrene toxicity. Experiments with strains of mice engineered to remove (knockout) CYP1A1 and CYP1B1 reveal that CYP1A1 primarily acts to protect mammals from low doses of BaP, and that removing this protection accumulates large concentrations of BaP. Unless CYP1B1 is also knocked out, toxicity results from the bioactivation of BaP to benzo[a]pyrene -7,8-dihydrodiol-9,10-epoxide, the ultimate toxic compound," Ref:
https://en.wikipedia.org/wiki/Benzo(a)pyrene#Carcinogenicity
So, from the first paragraph above ("BaP was shown to cause genetic damage in lung cells..."), I gather that tobacco is definitely bad.
However from the second two paragraphs, it seems like the CYP enzymes 1A1 and 1B1 can actually have a
protective effect for
low doses of benzo[a]pyrene like what would be in cooked meat, and also that food you digest goes through more thorough detoxification than inhaled smoke. In other words, for trace amounts of BaP, kava inducing 1A1 might actually help detoxify the stuff...
So, I still don't really know, but all of the above suggests to me the following answer to your question (which keep in mind is just a semi-informed guess)..
Burgers + kava = unknown, but probably OK. (And also don't smoke tobacco, which I'm sure you already know...)