Bath's late surge was not enough to avoid Champions Cup defeat in the La Rochelle fog, leaving them out of the last-16 spots with one game left.
It looked as if things could have been much worse than the eventual 39-21 scoreline when Ronan O'Gara's La Rochelle led 20-0 after 23 minutes.
A yellow card for Bath flanker Tom Ellis did not improve matters and the hosts scored three more tries.
But Bath responded with three of their own in six minutes.
After two Pierre Popelin penalties, returning centre Jonathan Danty cut down the left to score La Rochelle's opening try.
Remi Picquette followed suit and, with Ellis in the sin-bin, the lock added a second in the corner.
Dillyn Leyds secured the bonus point for La Rochelle off a line-out and an impressive run by Raymond Rhule ended in the wing scoring the hosts' fifth and final try.
Shortly after, Bath fought back as Johannes Jonker crossed first and Joe Simpson went over in the corner three minutes later.
The visitors continued to find some flow in attack and Ruaridh McConnochie's score with 17 minutes remaining offered the slightest hint of a comeback, but the Premiership strugglers were kept quiet for the rest of the evening.
Line-ups
La Rochelle: Dulin; Leyds, Sinzelle, Danty, Rhule; Popelin, Kerr-Barlow; Priso, Bourgarit, Atonio, Sazy, Picquette, Paul Boudehent, Bourdeau, Alldritt (capt).
Replacements: Bosch, Aouf, Papidze, Lavault, Tanguy, Le Bail, Plisson, Pierre Boudehent.
Bath: McConnochie; Prydie, Joseph, Clark, Muir; Redpath, Spencer; Schoeman, Doughty, Rae, McNally, Ewels (capt), Ellis, Richards, Bayliss.
Replacements: Du Toit, Morozov, Jonker, Spencer, Williams, Simpson, Butt, Hamer-Webb.
Match officials
Referee: Craig Evans (Wales)
Assistant referees: Elgan Williams (Wales) and Rhys Jones (Wales)
TMO: Ben Whitehouse (Wales)