What is the role of various additives in a titration of vitamin C with N-bromosuccinimide
Thanks, Martin, for pointing out the relevant paper with all the answers!
Basic procedure
This titration was first described by Mohamed Zakibarakat, Mohamed Fathy Abd El-Wahab, and Mohamed Mahmoudel-Sadrin in 1954 (doi.org/10.1021/ac60100a013) as an alternative to using 2,6-dichlorophenol-indophenol as titrant. Here is the procedure for a solution of pure acsorbic acid:

Into a 50-m1. conical Erlenmeyer flask, a known volume of the unknown ascorbic acid solution is introduced — i.e., 1 ml. contains 1 mg. of ascorbic acid. Then 5 ml. of 4% potassium iodide solution, 2 ml. of 3% acetic acid, and 2 drops of starch solution as an indicator are added. The aqueous N-bromosuccinimide solution (0.1% weight per volume) is introduced into the microburet and is allowed to run drop by drop, into the ascorbic acid solution, with continuous shaking. The end point is reached when the last drop of the N-bromosuccinimide solution added produces a permanent blue color in the ascorbic acid solution.

For fresh fruit, there is an additional step to prepare the analyte:

Edible Fruits. The new method has been applied to the estimation of ascorbic acid in certain edible Egyptian fruits known to contain ascorbic acid—e.g., oranges, lemons, tomatoes, grapes, watermelons, and water cress. The fruits used in the estimation of ascorbic acid were almost ripe. The sampling and extraction of the material under examination must be carried out with the minimum delay, so that no significant change in ascorbic acid content takes place prior to analysis. Each sample is squeezed or bruised and the juice is directly received in a known volume of 20% trichloroacetic acid, so that the juice is diluted with the stabilizing acid at least twice, to avoid the oxidation of ascorbic acid and also to precipitate interfering substances such as protein and thereby facilitate subsequent clarification of the extract. The acidulated juice is fil-tered and the determination is carried out on a known volume of the clear filtrate as described previously. The N-bromo-succinimide solution used in the titration is 0.01%—i.e., 1 ml. contains 0.1 mg. of N-bromosuccinimide (see Tables XI and XII).

Oxalic acid
Note that they did not mention oxalic acid, but they did use fairly acidic conditions to inhibit redox enzymes and precipitate proteins.
Oxalic acid does not interfere in the determination:

Interfering Substances. The following substances which might interfere have no influence on the titration process: carbohydrates such as glucose, lactose, sucrose, and starch; diketogulonic acid; reductones and reductic acid; urea; uric acid and creatinine; alcohols; formaldehyde; acetone; ethyl acetate and acetoacetic ester; thiamine hydrochloride and ribo-flavine; oxalates, tartrates, and citrates; amino acids such as glycine, alanine, valine, and isoleucine; and ferrous and ferric salts. The only interfering substances that are also oxidized by N-bromosuccinimide before iodine is liberated from potassium iodide in acetic acid medium include sodium sulfite, sulfide, thiosulf ate, and thiourea.

Role of iodide
They nicely describe the two relevant redox reactions used in the procedure:

The fact that ascorbic acid reacts very rapidly with N-bromo-succinimide, whereas many of the interfering substances react more slowly or even do not react at all, provides a reliable titrimetric method for the determination of ascorbic acid. N-Bromosuccinimide is an oxidizing agent and thus can liberate iodine from potassium iodide in aqueous acetic acid medium, but it oxidizes ascorbic acid to dehydroascorbic acid preferentially. Until all the ascorbic acid present in the solution is oxidized, no iodine is liberated from potassium iodide.
The slightest excess of N-bromosuccinimide added, after all the ascorbic acid content has been oxidized, will liberate iodine from potassium iodide, which is easily detected by the blue color developed with a few drops of starch solution added at the beginning of the titration process. The end point is thus easily observed by the appearance of a blue color in the ascorbic acid solution or extract. The presence of other reducing substances does not interfere with the titration with N-bromosuccinimide, since iodine is selec-tively liberated from potassium iodide before reducing substances present, other than ascorbic acid, are oxidized by N-bromosuccinimide, and consequently the end point is easily determined in the presence of starch. The end point is definitely blue in the case of pure solutions and pharmaceuticals, but violet in the case of biologicals and fruits.

So it would seem that as long as there is sufficient iodide to give the color change for the endpoint, the exact amount does not change when the endpoint is observed.