Sign language can't be written with the alphabet, because sign language and spoken language are fundamentally different.
Spoken language consists of a series of distinct sounds made one after the other in a specific order. The Latin alphabet was designed to reflect how spoken language works, with the letters A to Z representing twenty-six distinct sounds, and the letters written in lines (like this one) to show the order that the sounds are made in.
Now compare this to sign language, which has distinct handshapes making specific movements in space, and facial expressions which are part of the language.
Not only does sign language have far more than twenty-six handshapes and facial expressions, but the movements in a sign often occur simultaneously. Movements and simultaneous action cannot be represented with the Latin alphabet – it can only show one thing happening after another.
For these reasons, SignWriting includes symbols which directly represent handshapes, facial expressions, movements, and simultaneous action. This is why the SignWriting alphabet is so different from the Latin alphabet.