Well, there is so-so news, and some bad news.
The bad news is that the records in many Clare parishes don't start early enough to cover baptisms in the 1830s. I don't see any matching baptisms for your two ancestors of interest using the index I have access to (Roots Ireland). However, Ancestry has a somewhat different set of parish records indexed, so some hope there.
There is also some hope of at least pinning down candidate localities. Both Michael and Bridget were born around the time of the tythe valuations, and their fathers were both farmers, so they should be included. I would look for the two fathers in localities not that far apart. That means checking actual locations on a map, as townlands in different parishes can be close, or even adjacent.
We can search for John Han* in the tythe records for Co. Clare with the following URL
http://titheapplotmentbooks.nationalarchives.ie/search/tab/results.jsp?surname=hann*&firstname=john&county=Clare&parish=&townland=&search=Searchand we get 9 results.
Taking as an example the three entries for John Hannan in Craglea townland, civil parish of Killaloe, the next step is to look at the original page:
http://titheapplotmentbooks.nationalarchives.ie/reels/tab//004625670/004625670_00087.pdfOne can see that there are really just two entries (that I can see anyway), for John senior and junior and one can also see the size of their farms. [In Irish acres, not English].
Of course, there are several other candidates also. RootsIreland does not have the baptismal records indexed for Killaloe, but Ancestry probably does.
Turning now to Thomas McInerney (note the customary capitalization of the I),
the following URL is a link for a search for Thomas McIner* in the Co. Clare tythe valuations:
http://titheapplotmentbooks.nationalarchives.ie/search/tab/results.jsp?surname=mciner*&firstname=thomas&county=Clare&parish=&townland=&search=SearchThis time there are 25 results.
One of them, is a Thomas McInerney in the townland of Clonfadda, civil parish of Killaloe.
Link for original page:
http://titheapplotmentbooks.nationalarchives.ie/reels/tab//004625670/004625670_00064.pdfSo, one example of two families with the right names, in close proximity, at the right time. Parish registers exist for Killaloe, and may rule these particular families out - I just don't have access to an index.
Of course, this is not definitive and I am sure you can find other such matches in the tythe valuations, but this exercise gives some idea of possibilities. If I were OP I would also pursue DNA matching, and see where matching families originated, and compare with tythe record locations.
Best of luck!
An important final thought for OP - allow for spelling variations, and abbreviations!
