I organized the trainers by area and if they can be rematched or not.
North Boulevard
Lumiose Gang Member Sedna (Can be battled once)
Lumiose Gang Member Eris (Can be battled once)
Lumiose Gang Member Nix (Can be battled once)
Suspicious Woman "???" (Can be battled once)
Suspicious Child "???" (Can be battled once)
Pokémon Trainer Malva (Can be battled in the Elite 4 numerously)
South Boulevard
Pokémon Professor Sycamore (Can be battled once)
Preschooler Natalie (Can be battled once)
Preschooler Lily (Can be battled once)
Schoolgirl Callie (Can be battled once)
Schoolboy Macon (Can be battled once)
Suspicious Lady "???" (Can be battled once)
Estival Avenue
Garçon Jacopo (Can be rematched)
Bleu Plaza
Owner Toro (Can be rematched)
Waitress Paget (Can be rematched)
Chef Kamaboko (Can be rematched)
Vert Plaza
Punk Girl Lillian (Can be rematched)
Beauty Aimee (Can be rematched)
Jaune Plaza
Beauty Anais (Can be rematched)
Hibernal Avenue
Black Belt Killian (Can be rematched)
Rouge Plaza
Butler Chalmers (Can be rematched)
Punk Guy Faust (Can be rematched)
Lumiose Gym (These trainers can be battled only once)
Schoolboy Arno
Schoolboy Sherlock
Schoolboy Finian
Rising Star Helene
Rising Star Nelly
Rising Star Estel
Ace Trainer Mathis
Ace Trainer Maxim
Ace Trainer Rico
Poké Fan Abigail
Poké Fan Tara
Poké Fan Lydie
Leader Clemont
Lysandre Café (These trainers can be battled only once)
Team Flare Grunt (Male)
Team Flare Grunt (Female)
Both disguised as waiters
Lumi Cabs (Can be battled every time you don't have enough poké-dollars)
Driver Caboot
Driver Carter
Driver Clarence
Driver Cormick
Driver Coupe
Driver Hackney
Driver Hansom
Driver Schaffer
Driver Wayne
Driver Weaver
Driver Wheeler
Looker Bureau
Essentia (Can be rematched)
Restaurant Le Yeah
Waitress Talia (Can be rematched)
Garçon Georges (Can be rematched)
Owner Yannick (Can be rematched)
Chef Albern (Can be rematched)
Restaurant Le Nah
Chef Roger (Can be rematched)
Restaurant Le Wow
Owner Christophe (Can be rematched)
Garçon Morris (Can be rematched)
Waitress Paula (Can be rematched)
Sushi High Roller
Furisode Girl Gwyneth (Can be rematched)
Furisode Girl Cliantha (Can be rematched)
Furisode Girl Thalassa (Can be rematched)
Furisode Girl Nerissa (Can be rematched)
Owner Kazu (Can be rematched)
Other Trainers: (Cannot be rematched)
AZ (During the parade where the player becomes champion)
Summary
Trainers that can be fought once: Lumiose Gang Member Sedna, Lumiose Gang Member Eris, Lumiose Gang Member Nix, Suspicious Woman "???", Suspicious Child "???", Pokémon Professor Sycamore, Preschooler Natalie, Preschooler Lily, Schoolgirl Callie, Schoolboy Macon, Suspicious Lady "???", Schoolboy Arno, Schoolboy Sherlock, Schoolboy Finian, Rising Star Helene, Rising Star Nelly, Rising Star Estel, Ace Trainer Mathis, Ace Trainer Maxim, Ace Trainer Rico, Poké Fan Abigail, Poké Fan Tara, Poké Fan Lydie, Leader Clemont, Team Flare Grunt (Male), Team Flare Grunt (Female), and AZ.
Trainers that can be rematched: Pokémon Trainer Malva, Garçon Jacopo, Owner Toro, Waitress Paget, Chef Kamaboko, Punk Girl Lillian, Beauty Aimee, Beauty Anais, Black Belt Killian, Butler Chalmers, Punk Guy Faust, All the Cab drivers, Essentia, and all the restaurants.
During the storyline of Lumiose City, you can battle restaurants, cab drivers, and trainers in alleyways of the city, in the plazas and avenues listed above, more than once. The trainers in the Looker Bureau Quest include all the Lumiose Gang Members, the "???" people, and all the preschoolers/ school-kids in the South Avenue alleys can be battled once. You can rematch Emma once per day, on Tuesday, Thursday, and Saturday.
A. Lot. Of. Trainers. In. Lumiose. Dang.
Let me know if I need to edit anything.
Source 1, Source 2, Source 3, Source 4, Source 5, Source 6, and Source 7.
