Ulster visit Wasps for their final Champions Cup pool game on Sunday, 21 January with the winners likely to secure the top spot in Pool One.
Bales opened the scoring with an early penalty after huge early pressure from La Rochelle had almost put Vincent Rattez in from a Jeremy Sinzelle cross-kick.
A yellow card for replacement wing Paul Jordaan saw La Rochelle reduced to 14 men and Ulster Rugby seized the initiative as Heineken Man of the Match Best forced his way over the whitewash from a rampant driving maul and all of a sudden the hosts turned the tables as the atmosphere lifted in Belfast.
Ulster took advantage of their numerical advantage and Christian Lealiifano - playing his final home game for Ulster - put a penalty into the corner.
Ulster got their noses in front five minutes before half-time.
Ulster went wide Louis Ludik's scoop pass allowed Stockdale to pick up and charge over in the right corner after swatting three defenders.
La Rochelle returned rejuvenated after an uncompromising interval talk from head coach Patrice Collazo and scrum-half Bales raced home a simple try after Victor Vito sliced open the Ulster defence and converted his own score as the tourists regained the advantage.
But Ulster immediately hit back when replacement Darren Cave latched on to a spilled ball in La Rochelle's 22 to put Timoney in space.
The number eight did well to drive over after 45 minutes.
But Ulster's failure to gain a maximum-point triumph, along with the French team managing a vital losing bonus point - thanks to a try, conversion and two penalties from scrum-half Alexi Bales - meant La Rochelle remained in the mix ahead of their home game against Harlequins.
The remainder of the game was mostly dominated by Ulster, but neither side managed any further scores.