Here's the thing about Qi staying out of the war- they had no choice. By the time Qin engaged in its final series of major offensives against the other states, the civil war had gone on for centuries. Qi had suffered major defeats by its closest neighbors. Any support it could provide to its neighbors would be mediocre. It didn't even have the resources to put up more than a token defense when Qin invaded it.
To envision a scenario in which Qi plays a major part in the closing battles of the warring states means we have to diverge much earlier in history, back to a point in the war when Qi was still powerful. Here's the relevant time frame as per Wikipedia:
The Period of Qi (301-284 BC)
300 BC: Lord Mengchang of Qi, grandson of the former King Wei of Qi came to power when King Min acceeded to the throne. He made a westward alliance with the States of Wei and Han. In the far west, Qin, which had been weakened by a succession struggle in 307, yielded to the new coalition and appointed Lord Mengchang its chief minister. This "horizontal" or east-west alliance might have secured peace except that it excluded the State of Zhao.
298 BC: Zhao offered Qin an alliance and Lord Mengchang was driven out of Qin. The remaining three allies, Qi, Wei and Han, attacked Qin, driving up the Yellow River below Shanxi to the Hangu Pass. After 3 years of fighting they took the pass and forced Qin to return territory to Han and Wei. They next inflicted major defeats on Yan and Chu. During the 5-year administration of Lord Menchang, Qi was the major power in China.
294 BC: Lord Mengchang was implicated in a coup d'etat and fled to Wei. His alliance system collapsed. Qi and Qin made a truce and pursued their own interests. Qi moved south against the State of Song whilst the Qin General Bai Qi pushed back eastward against a Han/Wei alliance, gaining victory at the Battle of Yique. In 288 BC the two rulers took the title of "Di", (帝 literally emperor), of the east and west respectively. They swore a covenant and started planning an attack on Zhao.
287 BC: The diplomat Su Qin, possibly an agent of Yan, persuaded King Min that the Zhao war would only benefit Qin. King Min agreed and formed a 'vertical' alliance with the other states against Qin. Qin backed off, abandoned the presumptuous title of "Di", and restored territory to Wei and Zhao. In 286 Qi annexed the state of Song.
285 BC: The success of Qi frightened the other states (some say that this was part of Su Qin's plan). Under the leadership of Lord Mengchang, who was exiled in Wei, Qin, Zhao, Wei and Yan formed an alliance. Yan had normally been a relatively weak ally of Qi and Qi feared little from this quarter. Yan's onslaught under general Yue Yi came as a devastating surprise. Simultaneously, the other allies attacked from the west. Chu declared itself an ally of Qi but contented itself with annexing some territory to its north. Qi's armies were destroyed and the King Min was slain. Qi was reduced to the two cities of Ju and Jimo. King Min himself was later captured and executed by his own followers.
After 284 BC Tian Dan was able to restore much of the state's territory, but it never regained the influence it had under King Min.
More specific information on the players involved in that time frame.
Lord Mengchang-
http://en.wikipedia.org/wiki/Lord_Mengchang_of_Qi
King Min-
http://en.wikipedia.org/wiki/King_Min_of_Qi
Su Qin-
http://en.wikipedia.org/wiki/Su_Qin
The way I see it, the two changes to the timeline need to be the death of Su Qin (or at the very least him being stationed elsewhere in the warring states) and King Min deciding the invasion of Song is unnecessary and imprudent. If there is no falling out with Lord Mengchang, King Min can hopefully be guided to more prudent courses of action and thus will not invite the wrath of every other state upon Qi. A stronger Qi can then check Qin's power more successfully. This also prevents the sack of the Jixia Academy, which in turn prevents the diaspora of Chinese scholars and thus Hungandi's exposure and adoption of Legalism as Qin's primary philosophy (though the militaristic Qin may adopt the philosophy anyway, it has considerable applications and appeal for a would-be conqueror).