Evaluation of Gut Microbiome in Patients With Prostate Cancer - Full Text View - ClinicalTrials.gov (I found an NIH writeup of the results and nothing striking was found)
Study to Detect Changes in Urinary and Gut Microbiome During Androgen Deprivation Therapy and Radiation Therapy in Patients With Prostate Cancer - Full Text View - ClinicalTrials.gov
I don't know if we have the answer to that. Zytiga and Metformin "might" increase some potentially beneficial bacteria. As pjoshua posted, standard ADT might increase some potentially harmful bacteria (Zytiga is an interesting exception - something to do with the mechanism of action).
I take Zytiga and had my gut biome measured with a biomesight.com/ test.
My beneficial bacteria are supposedly 20x the average. Verrucomicrobia phylum (includes akkermansia).
But I do not take Zytiga to change my gut biome. I take it to block cyp17 (standard therapeutic use).
Until we know more about the microbiome, it makes sense to me to eat plenty of fiber and maybe add some probiotics like the ones you mentioned. I do find it interesting that there are a few dozen RCTs in progress. Hopefully, this is an area where the smoke comes from a fire.
25. Akkermansia is speculated to be a PCa beneficial gut bacteria: Strategies to promote abundance of Akkermansia muciniphila, an emerging probiotic in the gut, evidence from dietary intervention studies - PMC
26. Akkermansia is speculated to be a PCa beneficial gut bacteria: Abiraterone acetate preferentially enriches for the gut commensal Akkermansia muciniphila in castrate-resistant prostate cancer patients | Nature Communications
Not really. I read about it in an article. It was only a couple hundred bucks so I didn't bother looking around. And I figured that the only way it would be actionable is if something was way off.
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UCLA is expanding access to the Ga-68-PSMA-11 PET scan
Berberine - One way to increase Akkermansia in your gut. Curcumin helps the gut microbiome as well. 
nanoparticles prostate cancer treatment?
Resveratrol Revisited. For Gut Microbiome.
The human microbiome and its link in prostate cancer risk and pathogenesis
