I am afraid I don't have the specialist facilities to trace the painter but I can tell you a little about the Wrestlers Inn, and indicate how you can learn more.
You say that the painting was done in 1881 and 1881 is a UK census year. If you visit the Ancestry web site and carry out a search (no name) looking for people who were living in "Aldenham" county "Hertford" with a street address "Wrestlers" you will get three names. They are Tom How (the publican), his wife and a servant. You may well be able to find the inn in other censuses (but it may not be so simple) and you should be able to find out more about Tom and his wife using the techniques described in the tutorial on this site.
Tom How may not have been there when the painting was done as in the 1882 Kelly's directory there is an entry for Aldenham which reads:
Halsey Wm. Wrestlers' Inn, Aldenham Wood
A number of other directories are available to you online at the Digital Library of Historical Directories and you may want to see who was the publican in other years.
There is no mention of the Wrestlers in Hertfordshire Inns & Public Houses and it may well be that, like many others, the Inn closed before the First World War.
Colin Meager writes: We corresponded about Jane Philbey who had the Green Man in Tring and indeed also about her brother Thomas Meager who had the Harrow in Akeman Street.
I was interested in your piece on the Wrestlers Arms at Aldenham as our family has a connection there too ! One of Jane's other brothers James Meager took on the lease of the Wrestlers Arms in Aldenham Wood in April 1867. James at that time was running the Boot Inn in Edgware but as his son David was about to marry Sarah Gosforth in Aldenham he very likely took the pub on for his son to run. David Meager married Sarah in June 1867 but unfortunately David died in December 1867. Sarah remained in Aldenham and is shown on the 1871 census back living with her family.
Yet another Hertfordshire pub that he have connections with ! Incidentally James also ran the Clarendon Arms at Chandlers Cross – he is there on the 1851 census and gave up the lease in the mid 1850’s.
Further to may earlier comments I note that in 1890 Mrs Elizabeth Catherine Dickens was the publican at the Wrestlers' Inn, but it was not listed in the 1895 Kelly's Directory. However in that year a Mrs Elizabeth Dickens did have a public house called the Battle Axes in Aldenham Road. Aldenham - which still exists as a pub restaurant. However it was not just a change of name as an on line search provides the following text:
The Aldenham estate in the 19th Century was one of the largest and longest in the county of Hertfordshire. It stretched from Aldenham Village in the east through Letchmore Heath and on past where Elstree Aerodrome now stands to the Aldenham reservoir. The land was owned at the time by Henry Hucks Gibbs, MP for the City of London who decided to reorganise the estate and closed the old Elstree Road and Grubs Lane. The Wrestlers pub was situated on the junction of the two roads and was demolished when the new roads Butterfly Lane and Aldenham Road were created. Mr Gibbs realised that his workers would still need a pub in the centre of the estate and built the Battle Axes on the new junction.
The remains of brick foundation walls were the only features/deposits revealed during the watching brief. The walls were possibly those of the Wrestlers Arms with its attached timber outbuildings, which was demolished in c.1891 as part of the major alterations undertaken at the time. The Hertfordshire Historic Environment Record located the public house as directly outside the north-east boundary of the site so such potential remains could be anticipated. An Ordnance Survey map of 1877 records the public house (‘The Wrestlers) as having stood at the road junction but did not indicate where on the junction it was located. The shallowness of the foundation walls may imply that they either belonged to the partly weather boarded public house, in which case they must have been more substantial, and/or that they belonged to the timber outbuildings. If the former, the building material was probably re-used perhaps for the construction of the Battleaxes public house at the corner of the new Butterfly Lane (i.e. the road that replaced Grubb Lane 180m to the north). A spread of Post-medieval material, c.15m wide, was recorded at the south-west end of the stripped temporary access road. It was presumed to have been derived from the demolition of the