TL;DR: Despite the syntax, which is a matter of style, this is essentially describing mixed feelings.

Moncrieff’s choice

Here are some excerpts from the TLF’s entry on fanfaron :

I. — Emploi adj. Qui affecte la bravoure; qui cherche à s’imposer par le verbe ou l’attitude en exagérant son mérite et son courage. (Quasi-)synon. bravache, hâbleur, vaniteux.

B. — [En parlant d’un attribut de la pers., d’un comportement, d’une réalisation] Un air fanfaron; une allure, une brusquerie fanfaronne. […] Mais il n’avait ni la galanterie facile ni l’audace fanfaronne des jours de belle assurance (Pergaud, De Goupil, 1910, p. 48). Il parlait sans s’occuper de moi et sur un ton fanfaron (Lacretelle, Silbermann, 1922, p. 12).

From this I gather that common translations such as "boastful, swaggering, blustering" (WordReference), "blowhard" (Linguee), "braggart" (Google Translate) are not quite what is wanted. In the first place, they seem to translate the senses of what TLF concedes are only quasi-synonyms: bravache, hâbleur, vaniteux. In the second place, those English translations are all highly pejorative, whereas one catches a whiff of approbation in Pergaud’s « audace fanfaronne des jours de belle assurance » ; excessive self-assurance in the young can be obnoxious, and at the same time not out of place.

In any case, when we pair any of those translations with timidité, we run into more serious trouble. There is a certain wry paradox in a « timidité honteuse et fanfaronne » — we can see that Proust himself considers them at odds since he writes elsewhere in the volume « il y avait plus de timidité que d’orgueil » — but to translate this tension by the oxymoronic "swaggering timidity", for example, would be ridiculous. Moncrieff has opted for "blustering", perhaps the least pejorative of the above terms (as well as faintly echoing "flustered"), and it works well with "timidity" because can it be used of impersonal forces like the wind.

A closer look at the scene

But since any translation chooses only one facet of a word to put on display, and you’re wondering what (else) Proust meant, let’s probe fanfaron a little more.

une admiration et une gaieté où je crus discerner un peu de timidité, d’une timidité honteuse et fanfaronne

The act in question is that of jumping on a « tremplin naturel » over the head of a seated vieillard. The girl admiring and enjoying the jumper is of her troupe; she may be admiring both the acrobatics and the audacity of the girl who jumped such that her feet "brushed" the old gentleman’s cap.

Here’s where the timidity comes in: she is timid about admiring or enjoying the jump. Presumably she would think twice about doing it herself, and is unsure even about showing or feeling gaiety or admiration. That part, I think, is pretty easy to understand — it’s the desire of a girl to laugh but suspecting that it might not be courteous, or prudent.

It’s this timidity that Proust describes as both honteuse and fanfaronne. As aCOSwt points out, technically the abstract timidity itself has these qualities. In that case honteuse would suggest "shameful ~ something to be ashamed of". It’s also quite natural, however, to read through the syntax and see them as the girl’s feelings about her timidity. Moncrieff clearly reads the latter in translating honteuse as "shamefaced", which seems to describe physical embarrassment or blushing in the girl. Grieve collapses timidité and honteuse into "bashfulness", a quality of the girl, and suggests this direction also by "bare-face", which brings to mind another visible image. Under this light the paradox is stark: she is both shamefaced and bare-face!

Given these translators’ choices, I think it would be best to read honteuse and fanfaronne as attitudes of the girl regarding her timidity, or mixed with her timidity, rather than applying abstractions to abstractions. This, I think, is the beginning of clarity on the question.

The girl is thus honteuse; I think it’s best to read this as recapitulating timidité. That is, it’s not a shameful timidity — she is not ashamed of her timidity — but a timidity born of shame. She feels compunction about admiring and enjoying the jump.

On the other hand, she is also fanfaronne. Now again we can read this as either an arrogant timidity — a timidity of which she would boast — or we can read it as a feeling mixed with the timidity. In my opinion, the former involves a convoluted train of thought. I think it’s the latter, which fits neatly with what we easily understood earlier. In showing her admiration and gaiety she is proud and audacious (or she is vicariously proud and audacious through her friend).

So picture the girl looking on as her friend jumps. The feet brush the old man’s cap — she laughs and she watches the friend for a second with admiration. She feels she shouldn’t take part with her whole heart, that she might be breaking a taboo to do so, so it’s not quite as loud or free as it might be. But she also feels, in the company of the other girls, a certain licence, maybe even a right, to thumb her nose at taboo, to take liberties and enjoy them.

Have you ever laughed when you knew you shouldn’t, and simultaneously half-hid it and felt freer — or transitioned quickly from one to the other? I have!

Luke Sawczak’s response is very thoughtful, thought-provoking and thorough. The ambiguity of shameful is worth analysing (assuming the same ambiguity exists in French honteuse); it can mean, without recourse to reattribution, that the girl’s timidity either made her feel shame or was in itself a shameful manifestation. However, I think to displace the adjective fanfaronne from the noun timidité, of which it is clearly an attribute, does not do Proust justice. It would be grammatically more correct and makes more sense to me to translate it as overt or showy. In other words, she really put the timidity act on. This in turn means that honteuse indicates that her timidity, being an act (at least in part), was shameful in itself (from a moral perspective).