Diplodicus, 'Leman' (or Leaman in the 1841, where the difference between the enumerator's 'L' & 'S' is clearer) is certainly the correct reading - he's clearly the Elijah [sic] Leman, son of Thomas Leman & Phebe Hunter, baptised with his brother William at Fressingfield on 25 Sep 1808. You'll already have found all this, I'm sure, but Thomas married Phoebe Hunton at Metfield on 13 Aug 1805 - his age in the '51 Census suggests a birth year of c1769, that in the '41 as c1771-76, but ages of the elderly were of course very shaky in those days (and the '51 didn't know his birthplace, so I'd take the age of 81 with a pinch of salt). It's interesting that in 1841 father Thomas is spelt 'Lemon', while son Elisha is 'Leaman' - presumably they pronounced it differently, and the enumerator wrote what he heard.
The '41 says he was born in the county, so despite the distance (30+ miles) I would have said he could be the Thomas Leman, son of John & Mary, baptised at Semer on 4 June 1775. However that Thomas married someone else (Sarah Hill) in 1795, and although the last of their children was baptised before Thomas & Phoebe married, another source suggests his wife Sarah did not die until 1810, which must rule him out. And besides, although Thomas of Semer's forbears are traceable back several generations, they came from Lavenham...which is really in the wrong direction if we're trying to find a link with the Beccles family!
So to get (finally) to your question, there seem to be plenty of Lemans/Lemons in many parts of Suffolk. Sir John's family were in the Beccles area in the early C16th, and probably long before. Beccles is only 12 or 15 miles from Metfield/Fressingfield; and although I fear you'll never prove it, I would frankly be be surprised if your Lemans were NOT distant cousins of the posh lot! But you may never know whether they branched off before the other family clawed their way up in the world, or are descended from a junior branch that gradually slipped back down. Downward social mobility must be quite as common as the upward variety, but of course it is seldom recorded - one occasionally gets reminded of this mathematical necessity when an Australian bus driver unexpectedly inherits an earldom from a distant relation...and a former Garter King of Arms, Sir Anthony Wagner, convincingly calculated that in the early 1960s there had to be around 2 million living descendants of Edward III!!