Chatham Dockyard was one of the main Royal Navy Dockyards. As a rope-maker, it's certainly possible that he was working there.
You might want to look at Records of Royal Naval dockyard staff to see if Thomas appears in those records. That might lead to further clues for research.
For example, the pay books for Chatham Rope Yard, covering the years 1683 to 1865, are in the record series ADM 42/287. That series has not been digitised, so you would need to either visit Kew in person, or employ someone to go there on your behalf.
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If you can't get to London in person, it may be worth taking a look at the Random Acts of Genealogical Kindness website to see if there's someone who would be able to check the records for you.