The Tigers needed all four league points to retain hopes of clearing the Bears in the final table, but instead they completed the season without a win from their final eleven fixtures despite an outstanding display from Charles Wright.
Their task was made tougher by the pre-meeting withdrawal of captain Kyle Howarth, who aggravated a shoulder injury racing at Somerset on Wednesday, but they did take an early lead on Teesside as Broc Nicol won Heat 2 and home man Dimitri Berge crashed out of the next race.
But the Bears went ahead for the first time with a 4-2 in Heat 4, before taking their first 5-1 two races later courtesy of Thomas Jorgensen and Nika Lunna.
Tigers profited from Ben Barker's engine failure in Heat 6, but they soon found themselves eleven points down as Kasper Andersen, who won his second outing, had no answer to Berge and Jonas B Andersen in Heat 9.
To their credit, the visitors mounted a response with a 4-2 in Heat 11 led by Wright, and the Tigers' No.1 was at it again in the next when he replaced Connor Mountain as a tactical substitute and partnered Andersen to a 5-1 to slash the gap to five.
Wright made it a hat-trick in a shared Heat 13, but hopes of a dramatic comeback victory ended in the next as Andersen took the flag even though Nicol and Todd Kurtz both passed Coty Garcia.
And a fourth successive Wright victory, re-passing Jorgensen along the way, wrapped things up in Heat 15 to at least secure a league point for Tigers '“ ahead of what will surely be a major winter re-build.
REDCAR 47: Thomas Jorgensen 10, Dimitri Berge 8+1, Jonas B Andersen 8+1, Nike Lunna 7+1, Ben Barker 6+1, Fernando Garcia 4, Jordan Stewart 4. SHEFFIELD 42: Charles Wright 17+1, Kasper Andersen 8+1, James Shanes 6+1, Todd Kurtz 5+1, Broc Nicol 5, Connor Mountain 1, Kyle Howarth r/r.