I have written about this in the past:
- A mouth full – for better or worst 
- Oral Probiotics 
- Your mouth can trigger flares 
- Science based mouth wash for ME/CFS 
- Probiotics for mouth, sinus and eyes 
When I first started Microbiome Prescription the main and most popular provider was uBiome (dearly departed) which explicitly offered oral microbiome analysis. I had about a dozen uploads. Eventually I stopped support because there was not going to be sufficient uploads, even if I was waited for a decade.
Stomach Acid does not nuke probiotics!
This is a common internet legend which disagrees with both studies and common sense!!!
Common Sense Anyone?
This is the key question: “If you have a bacteria in your gut, how did it get there?”
Assuming that you do not believe that bacteria has perfected teleportation, then there is just one route: Through the mouth, the stomach, etc. It is the great trek!
Studies Supporting No or Little Impact
There are two German probiotic that are suspended in water, added to water and drunk. These studies indicate that they do survive! In fact, they persist!
- Persists: Symbioflor-2 E.Coli
- Symbioflor-1  Enterococcus faecalis
Looking at Vet Practices, adding probiotic to food or water is well establish. Some sites actually advocates opening capsules. Custom Probiotics advocates all of their probiotics be taken in a glass of water.
Where does this myth come from?
I strongly suspect marketing — if you compare the cost per BCFU from Custom Probiotics to your usual health food store probiotic capsules you will see the costs can be 10x higher in the capsules That is NOT the cost of putting them into capsules. In marketing, claims about the importance of capsules is differentiate product to get you to buy brand X and not Y.
People are also willing to pay for convenience. Buying flour to bake a load of bread is much cheaper than buying a ready made load of bread. At our local coop, a custom loaf of bread was selling for $11.95 — people are lazy.
I also suspect that stomach acid eliminating probiotic has been used as an excuse by many vendors. The likely cause of probiotics not staying around is where they were sourced: Human or Animal. “In general, their optimal growth temperature ranges between 36–38°C and 41–43°C for human and animal origin strains,”  A probiotic bacteria is unlikely to thrive at a temperature 7°C (or 12°F) from it’s preferred temperature. It will likely reproduce less and be less robust (allowing other bacteria to beat it up).
This comes back to my old soapbox: Buy Only Probiotics where the STRAIN (and not just the species!) is listed – and that specific strain has been researched (Ideally for the condition that concerns you. Use this link to look up most, for example periodontal disease), and that the origin is human. I would suggest constantly emailing the manufacturers!!! It is likely the only way that the situation will improve.
Oral Microbiome Is associated with many conditions
- ” In conclusion, this study suggests significant associations of the Oral Microbiome diversity with certain mental health dimensions such as depressive symptoms and anxiety.”  
- Oral Microbiota, Its Equilibrium and Implications in the Pathophysiology of Human Diseases: A Systematic Review. 
- Profiling the oral microbiomes in patients with Alzheimer‘s disease.
- Unlocking Modifiable Risk Factors for Alzheimer‘s Disease: Does the Oral Microbiome Hold Some of the Keys? 
- Oral and intestinal dysbiosis in Parkinson‘s disease. 
- Oral and gut dysbiosis leads to functional alterations in Parkinson‘s disease. 
- Changes in Oral Microbial Diversity in a Piglet Model of Traumatic Brain Injury. 
- Fungal Infections Of The Oral Mucosa. 
- Implications of oral streptococcal bacteriophages in autism spectrum disorder.
- Oral Microbiota Changes Contribute to Autism Spectrum Disorder in Mice.
My impression is that any condition with a neurological component (i.e. brain fog, impulse control, etc) is likely to have ORAL microbiome dysfunction.
There are many products available, for example, see this list. They will likely contain one or more of the following. For a bigger list, see researched probiotics here: mouth, sinus and oral.
- Bacillus Coagulans
- BLIS K12,
- BLIS M18,
- lactobacillus acidophilus,
- lactobacillus reuteri,
- lactobacillus paracasei,
- lactobacillus salivarius,
- salivarius thermophilus
- Symbioflor-1  Enterococcus faecalis
The “salivarius” indicate where it was first identified (mouth saliva). So it is normally in the mouth.
The key is for the probiotic to stay in the mouth for sufficient time to dislodge some other residents. To me, this appears to suggest:
- Brush and rinse after every meal
- You could do things like break apart an Oregano Oil capsule or drop Monolaurin flakes into the mouth and hold them there for as long as you can tolerate them…
- Take one or more lozenge afterwards… I would suggest taking one at a time, then perhaps change to a different one when dissolved. There are a few probiotics that are available as pressed pills. If the taste is not too bad, I personally use Miyarisan(jp) [clostridium butyricum] and shin biofermin (jp) [Bifidobacterium bifidum, Enterococcus faecalis, Lactobacillus acidophilus] in this manner. Some others to consider:
- BioGaia Prodentis Mint Lozenges [Lactobacillus Reuteri]
- Flora, Super 5 Probiotic Lozenges [Lactobacillus Acidophilus: 60% Bifidobacterium Bifidum: 15% Lactobacillus Bulgaricus: 15% Streptococcus Thermophilus: 5% Lactobacillus Salivarius: 5%]
- NOW Supplements, OralBiotic™, [Streptococcus salivarius BLIS K12]
The ideal would be to take an oral microbiome test that reports percentile ranking of each bacteria (against other oral samples). I do not know of any one providing that. uBiome likely had that data, but they are no more.