The problem in trying to transcribe this piece of French language is that it might not be customary at all and so puzzle your correspondent or cause a misinterpretation of your intent. In any case I don’t believe that you can “go all the way” in English; it seems to me it would be too strange.

Although the particular combination I suggest here is very probably not used at all in the place you intend for it, its terms can be found in other parts of a letter; the only inconvenient is the transgression of customs, accepted ways, and on the ground of the adage “when in doubt leave it out” I wouldn’t use that type of ending without the assurance of causing myself no wrong (specially in a letter of application for a job).

Hoping my application will be considered favourably, I extend to you my best regards.

Truly yours,