Reform pronouncation instead
As several other posters have pointed out, the same word is pronounced differently in different parts of England (and even more so if we include other English-speaking countries), so settling on a phonetic spelling would require a choice of which pronunciation is "correct". Other people remarked on the horrific cost of converting written documents to a new spelling standard.
But there is a simple solution: Keep the spelling but change the "official" pronunciation to follow the spelling in a consistent way. Simply define a short list of letter combinations and single letters and what their "proper" pronunciation is, using the general rule that letter combinations take precedence over single letters so, for example, the h in "that" and "chat" is pronounced differently from the h in "hat".
My guess is that you would need only about 25 different letter groups to be special cased, which should be reasonable to expect users of the language to learn. Most common words would be pronounced close to how they are pronounced today (in some part of the world, at least), but some of the more irregular word like "women" and "infinite"/"finite" (one or the other) would change "official" pronunciation.
With a modified pronunciation, reading old texts is no problem, but people who have learned only the new pronunciation would have difficulty understanding old recordings of speech. But how often do you listen to old recordings or see movies that are more than a few decades old?
And, in any case, I don't think understanding "old" English would be any harder than understanding many of the dialects already out there.