Unfortunately there is no single ideal test for checking gut health.
I agree that hydrogen and methane breath tests can be useful for detecting SIBO, though be aware that there is disputation about what means what.
This however is concerned only with the small intestine, not the colon where the vast bulk of the gut microbiota live.
There are two types of test looking at colonic microbial content, the older complete digestive stool analysis (CDSA) based on culture of microorganisms, and the newer DNA based analysis.
A CDSA is not what you want. There are some parameters in the test which are useful - eg analysis of SCFAs, pancreatic enzyme production, fat and fibre content of stool etc, but the microbial analysis by culture which is the bulk of the test is useless. The cost is too great to justify the useful info.
There are two types of DNA based tests. One has specific targets - ie it detects only the organisms specified, while the other attempts to sequence all DNA present and so identify all organisms.
Here and
here are posts which talks more about these issues.
As for which tests to order - well of course it depends what you most want to know and how much money you have to spend. The Diagnostic Solutions GI MAP test looks like a sensitive and fairly comprehensive test if you want to focus on parasites and problematic bacterial species, but it doesn't give the full gut picture.
I've not had this test so am not sure if you can order it yourself.
The uBiome Explorer test is a fairly inexpensive way (about USD 100) to get a comprehensive picture of gut bacteria, which are far and away the most dominant gut inhabitants. It can also detect archaea those these are very minor in comparison with bacteria. American Gut and RedLabs Belgium offer similar tests though the RedLabs test must be ordered by a doctor and is considerably more expensive.
This is a citizen science project so you order the test yourself, you get the results and it is up to you to interpret them.
Very recently they have offered Explorer Plus - the same test plus eukaryotes (fungi, parasites etc). This is considerably more expensive and they haven't actually sent out the first kits yet. American Gut has offered such a test for some time but it is extraordinarily expensive.