Hello Allan, welcome back, start with a bit of useless information, was looking in McLysaghts Surnames of Ireland which says Carton is an English toponmymic mainly found in Ulster, while Cartan is from the Irish christian name Art.
The piece about him in the colourful obituary as a county constable is interesting, it is not a term I am familiar with, there was an Irish Contabulary at this point, later the Royal Irish Constabulary, there was also an Irish Revenue Police, disbanded in 1852 and interegrated into the Irish Constabulary.
If he was in the constabulary and posted in Meath and the areas mentioned except Navan are all close to Drogheda, indeed also Stamullen where we located some Cartons in the earlier part of your query. The constabulary were never posted in their home counties indicating he was born elsewhere.
It says his wife was Scottish too so a look at the Scottish people website (subscription) may assist if he married there.
I searched Roots Ireland for Carton and automatically varients, and got 92 births between 1796 and 1806, no Michaels, one Matthew in 1796 in Antrim, one no Christian name in Granard, Longford, mother McVitty and I see another in Louth but not a name given by yourself.
Surname spelling not correct.
Name: Matthew Mccartin Date of Birth:
Date of Baptism: 06-Jan-1796
Address: Parish/District: SHANKILL ST ANNE'S, BELFAST
Gender: Male County Co. Antrim
Denomination: Church of Ireland
Father: Matthew Mccartin Mother: Mary Mccartin
When searchng church records it is important to know what denomination you are searching.
If he was in the constabulary a record of his joining and where from may exist, including a letter of recomendation from a dignitary or some such, to my knowledge what ecords exist are held at Kew in London, England, however there exists a very good and extremely well moderated Facebook page 1816 to 1922 Royal Irish Constabulary - A Forgotten Irish Policeforce, assume you are on FB so find the page and apply to join and put your query in the mix, there is a published history of the police by Jim Herlihy and another on the Irish Revenue Police by the same man, he is involved in the page, there is also a publication of all the members, just names and serial numbers and an email to the reference section of Louth or Meath libraries should elicit the information on the number which will assist in research unless a county constable is something else.
You can see where the various townlands in Meath are here https://www.townlands.ie/meath/ you can also of course search for Louth on the site, Navan is the main town in Meath, less than 20 miles from the other areas you mention.
As subscription sites go, Roots Ireland has a larger collection of church records not found on other sites but it is not a complete set either, I think you should initially follow the police trail to see if you can locate a birth place or place or origin and see where that takes you. Also you may find some record for him in court reports of the Petty Sessions, Find My Past has some Irish newspapers but the largest selection pre 1921 are on the British Newspaper Archive, I find some county library sites can give a very short synopsis of a report when you search their newspaper archive, not sure all counties have the feature though, one I am familiar with has them back to the 1860s so may be tricky getting earlier ones.
Let me know if you want links to various county library or archives.
Good Luck,
Pat