Can I use Sir William Brinton's ability immediately after discovering I have to lose Stamina?
You can use powers like that before drawing the encounter, or after resolving it (if you are not yet unconscious/insane), but you cannot use it during the resolution of an encounter.

From the Arkham Horror FAQ (page 14):


  Q: When can a character perform actions during a phase?
  For example, if a player with the ally Duke (“discard to
  immediately restore your Sanity to its maximum”) draws
  an encounter and has to roll a die and lose that much
  Sanity, when can he discard Duke to restore Sanity: not at
  all once the encounter card is drawn, before the roll, or
  after the roll?
  
  A: He may discard Duke after the roll, provided he
  has not been driven insane. Actions can be performed
  at any time during their listed phase as long as any given
  conditions are met. However, once an encounter is drawn,
  it must be resolved before any more actions can be performed.
  So, in the case of Duke, you can use him before
  or after drawing an encounter card, but if you draw the
  encounter card and it inflicts Sanity loss that reduces you
  to 0 Sanity, you go insane before you can use Duke. Flesh
  Ward is one exception to this rule because it prevents a
  Stamina loss and is intended to be used in response to such
  events. The Premonition Spell [CotDP] is another exception.