What is the most compact way to calculate and then Update a maximum value for each day's data?
I am not entirely sure but if the aggregated results are supposed to be per date and the date is production.scheduled_pull_date, then perhaps this:

UPDATE
  pOut
SET
  pOut.lead_ppm = (
    SELECT
      MAX(p1.lead_ppm)
    FROM
      dbo.production AS p1
    WHERE
       (p1.tank = 'B' or p1.tank = 'C')
       AND p1.scheduled_pull_date = pOut.scheduled_pull_date
  ),
  pOut.sulfur_ppm = (
    SELECT
      MAX(p2.sulfur_ppm)
    FROM
      dbo.production AS p2
    WHERE
      (p2.tank = 'B' or p2.tank = 'C')
      AND p2.scheduled_pull_date = pOut.scheduled_pull_date
  )
FROM
  dbo.production AS pOut,
  dbo.assay_temp AS tOut
WHERE
  pOut.tank = 'S'
  AND pOut.cell_num = 1
  AND pOut.scheduled_pull_date = tOut.formatted_date
;


Although I would also rewrite the join using the contemporary syntax:

FROM
  dbo.production AS pOut
  INNER JOIN dbo.assay_temp AS tOut ON pOut.scheduled_pull_date = tOut.formatted_date
WHERE
  pOut.tank = 'S'
  AND pOut.cell_num = 1


The assay_temp doesn't seem needed in the subqueries at all, and instead of matching scheduled_pull_date against formatted_date the subqueries should correlate with the outer query. Since you want aggregations per date, it seemed to me the subqueries should match against pOut.scheduled_pull_date.