I followed up on the idea that suggested that the death took place nearer Beaulieu than Minstead.
I found this
https://beaulieuhistorysociety.org.uk/images/newsletters/BHSNewsletter03Final.pdf which includes a summary of the research by Arthur Lloyd, a local historian
He cites several strands of evidence. These are the few I've been able to check
1) The Waverley Abbey Chronicles say that King John built the Cistercian Abbey of Beaulieu Abbey (Bellum Locum) close to the place where King William Ruffus was slain. (the Abbey was built in about 1204)
Rex Johannes quandam abbatiam de ordine cisterciensi construxit, quod bellum locum nominavit, propre locum ubi Willemus rex Ruffus occisus est.
2) Two much later writers give similar place names
Leland,(1502-1552) writes 'the place where it is sayde that Tyrelle kyllyd King William Rufus ys caullid Thorougham, and there standeth yet a chapel'
Stow,1525-650 writes,that William was hunting in' the New Forest of Hampshire,in a place called Chorengham,whence since a chapel was builded' (Lloyd suggests that the C is in error for a T)
Thorougham, no longer exists, it was, writes Lloyd, the later name of one of several manors called Truham and Trucham mentioned in the Domesday survey, These stood on the banks of the River Otter (from 1205 the River Beaulieu)
The National Archives website also stages that Truham/Trucham was 'in' Beaulieu
'Througham, In Beaulieu, New Forest, Hampshire, Folio 51 R .Domesday place name Truham/Trucham http://discovery.nationalarchives.gov.uk/details/r/D7300843
I think that there is no question that the death place should be changed to New Forest, Hampshire . Perhaps, changing it to Beaulieu,New Forest, might be a bit radical, since there is no certainty . (and it would presumably have been called either Truham or Trucham then )Nevertheless, the tradition goes back to whenever that bit of the Waverley Annals was written (the preface seems to suggest that the bit from 1157 onwards was near contemporaneous, so it would have been written about 100 years after William's death )
Edit, have been bold and changed death place to New Forest, Hampshire, written a paragraph with the details above.
If its a nice day, the Sir Walter Tyrell looks to be an excellent place to pickle ones holes