Like ideographic languages
Simply put, you're asking to pair an audible language to a written representation that has almost no meaningful direct correspondence.
That almost certainly means that, as in Jeff's answer, there will be a writing system more suited for directly representing the audible language.
However, if having a written language that uses Latin script is a specific goal... it's not inconceivable. We already have languages like that. For example, 冬 is pronounced "fuyu"¹, but the phonetic pronunciation has essentially nothing to do with the ideograph. Perhaps your Latin representation of this language similarly uses combinations of Latin characters that map to words, but lack any direct correspondence to pronunciations.
(¹ In Japanese. In some cases.)
Binary data can be represented in Latin script using various encoding processes. I could imagine a whistled language being encoded similarly; for example, certain letters correspond to specific starting pitches, others to ascending or descending scales, others (and/or symbols) to various clicks or other stops.
A "word" might look like «Bggh'ijf», indicating an absolute middle-low starting pitch ('B'; a lower-case letter would indicate a starting pitch relative to the previously voiced phoneme) which descends a sixth (each 'g' represents a third, and 'h' represents a step), followed by a click (apostrophe), followed by the previous note briefly sustained (each 'i' represents a duration of sustain) before rising a step ('j' represents a step up) and finally descending an octave ('f').
Of course, I'm assuming something like a chromatic scale with no minor intervals, which would actually break down almost immediately, but hopefully you get the general idea. In order to design a system like this that's actually usable, you'd need a solid understanding of how your audible language actually works, at least in terms of knowing its set of morphemes. Essentially, what you're doing is making a semi-arbitrary mapping of those morphemes onto Latin symbols.
The key point in either case is that you can get a mapping that uses Latin symbols, but obviously the "pronunciation" won't map from the Latin script in a way that anyone not exposed to the alien language will recognize.
If you take the latter approach (morpheme encoding, as opposed to arbitrary word mapping), there actually is not any particular reason to use non-Latin symbols, especially if the ability to interact with text using existing human equipment (like a US keyboard) is a goal. After all, most languages, including Latin-derived ones, don't have any particular rhyme or reason behind what symbols are associated with which morphemes.
That said, there is a certain amount of hubris in assuming that Latin scripts even still exist, but that can be explained away by translation convention.