Hi Ronnynel,
There are some 28 or more Roman Catholic Churches in Birmingham.
However it appears there were only two in 1832.
St Chad's , Queensway, Birmingham , B4 -6EU
and
St Peter's Broad Street, Birmingham ( now demolished )
Birmingham Library Archives should have records of St Peter's Broad Street ( Note address -more than one St Peter's )
Perhaps worth contacting them.
I have done a basic search of Transcribed records of St Chads , for the surname Thompson only.
We have to remember that many were illiterate during that period and that there are a number of variations of the surname.
Also that errors can occur in transcriptions.
Try searching 1841 census for Birmingham Warwickshire noting others on the page with the same surname ( 1841 census does not reveal relationships and age can be round off to nearest five years )
Then cross reference with 1851 Census and see if you can find the same family names with your John Thompson missing due to being tried in 1850.
Bo