Nashe gives various clues about his friend Beeston:
Age: Beeston has a "round cap" and "dudgeon dagger". By this period round caps seem to have been a fashion only among older men. (I remember seeing a reference in a letter by Robert Cecil to his father saying that since Sir Christopher Hatton had been promoted to the Privy Council he had taken to wearing a round cap, similar to the one Lord Burghley himself wore). The dudgeon dagger too seems to have been an item sported only by your man of mature years; it was apparently a false dagger of wood, painted and varnished to look like the real thing.
Status: Beeston has thrown away money on alchemy. He therefore is, or has been, rich enough to have money to invest. Nashe mentions no trade in connection with him but refers instead to Beeston's frequenting legal circles: "You are amongst graue Doctors, and men of iudgement in both Lawes euerie daie..." If Beeston actually had legal training of any kind he would have too much gravitas to be the dedicatee of this epistle; instead, this reference suggests a man with one or more suits at law. Later on, Nashe refers to Beeston's valuing "Iohn Davies soul". This presumably refers to Davies' poem 'Nosce teipsum', which at this time hadn't been published; if Beeston saw it in manuscript he had social connections with gentlemen at the Inns of Court.
This is a long shot, but in his Shakespeare v. Shallow Leslie Hotson refers to a William Beeston who was a landowner in Streatham. Hotson mentions him in connection with documents relating to William Gardiner, a corrupt justice of the peace. (See below for more information on Gardiner.)
The first document Hotson mentions is dated October 1556 and is a record of Southwark Court Leets. It shows Gardiner was fined three shillings and four pence for making an affray and shedding Beeston's blood. Gardiner was in his early twenties at this time; if Beeston was the same age group, then he'd be about 60 in 1589, when Nashe first knew him.
Beeston and Gardiner may have made up after their affray (was it drink-connected?) as there are two other documents relating to a sale of land by Gardiner to a William Beeston four years later. This William Beeston is "of Streatham, Surrey, gentleman" and has a wife called Joan. A later document refers to a Cuthbert Beeston, trying to recover £10 Gardiner owed him over a land sale in 1567.
So whether or not the Beeston whose blood Gardiner shed was the right one, there were obviously landowning gentry called Beeston south of the river. Landowners necessarily had connections with lawyers, so perhaps that's where Beeston's acquaintance with members of the legal profession comes in. Lastly, Streatham isn't very far from Lambeth, where Master Vaux who lost the rhyming contest came from. Case rests.