Plenty of reports about the 1833 court case most of which mention "... William Bingham, the son of Major Bingham, who can never be his heir-at-law... ".
An exception is The Roscommon Journal & Western Imperial Reporter which mentions "... Major Bingham and his natural son Wm ...". It also mentions "... Major Bingham is a Protestant, William Bingham is a Protestant ...".
Out of interest is this the Major Bingham that fought a duel with a Mr O'Donel in 1801?