I'm no expert, and I bet this has been explored, but I always felt that there's fantastic precedent in the world for using the Latin alphabet for writing sounds that occur in English; especially in the Scandinavian languages.
Eg the word "sauce", which consists of two identical consonants and a single vowel. Why not choose å for that vowel? Sås makes a lot of sense, and accidentally is also the Swedish word for "sauce" iirc. Or what about the two "th" sounds? Icelandic has got you covered: Instead of "this" and "thin", can't we write "ðis" and "þin"?
Note: this is not without precedent. The Norwegians totally overhauled their script until it was pretty much phonetic. This gave them fantastic benefits such as way fewer pupils struggling with dyslexia (at least, until learning Danish or English), and heartwarmingly lovely spellings of loan words such as "restaurang" and "stasjon".
It could be done.
Personally I think it's totally insane to use a phonetic script (as opposed to eg Chinese) but then not spell out the sounds. The Finns, Norwegians, Italians and Russians (and plenty others) got it right.