~ことになっている is for a custom or a decided rule, ~ことになる is something has been decided by someone other than yourself.
あすで1週間雨が降りつづくことになる。In the morning, it will have continually rained for a week. <<the situation has arisen; KOTO is a nominalizer, and it's now THINGになる, "becoming THING">>
>> あすで1週間雨が降りつづくようになる。This doesn't seem right to me. To my knowledge, the ようになる is used to mean "become able to do," usually as a gradual change and the subject is usually an animate being. The embedded clause, if it's affirmative, is USUALLY in the potential.
漢字が読めるようになりました。- I've reached the point where I can read kanji.
漢字を読まないようになりました。- I've reached the point where I can't read kanji. (can no longer read kanji)
漢字が読めるようになりません。- I haven't reached the point where I can read kanji yet. (haven't made it to that point)
As weather is NOT animate, your sentence cannot work.