If you just want to annotate sounds, it seems like your goals overlap a lot with some uses IPA is designed for. Wiktionary gives /sɑs/ as one pronunciation of sauce.

In any case, you'd have to solve the same problems people already using IPA for English deal with. For one, you'd probably want the same spelling system to be used across all major English dialects, at least necessitating something like using https://en.wikipedia.org/wiki/Diaphoneme.

An example of a situation that shows the necessity of this: https://www.reddit.com/r/linguistics/comments/2g50oo/working... (specifically the pronunciation of "bed", "bad" and "bid" in the GA and NZE dialects)

So for most speakers, different written symbols might be pronounced the same way (i.e. the map from symbol to sound is not injective, and furthermore the map changes depending on which dialect of English you speak).

Additionally, you probably can't simultaneously achieve the goals of

1. use only one character per phoneme

2. use only the 26 characters of the alphabet

If you insist on keeping property (2), some phonemes will be written using (say) two characters, but you'd have to make sure that there's no ambiguity (as there would be if there were three vowels written respectively "e", "i" and "ei").