According to Wikipedia, "Chet is a masculine given name, often a nickname for Chester, which means fortress or camp. It is an uncommon name of English origin, and originated as a surname to identify people from the city of Chester, England." JRRT would have known about this usage, one presumes, and so your supposition that the names of "Chetwood" and "Archet" are connected and should be pronounced accordingly makes a great deal of sense.
I could hazard a guess as to what those names are supposed to mean in Middle-earth but someone probably already knows and will jump in with the real answer
Most likely from Old English ar-cæt, meaning 'forest's edge'. As for pronunciation, in Old English, 'c' can be pronounced as 'k' (e.g. cyning = king) or 'ch' (cild = child). In this case, 'c' followed by 'æ', it would have been pronounced as 'ch'.