Why can't we say "sign in into"?

Sign in as a synonym for the verb access is not a separable verb. e. g. Send up, put down. He put down his boss. == He put his boss down. Sign in the guest and sign the guest in are equivalent.

Sign into the account and Sign the account into are not. the Guest is the direct object, whereas the account is not. Clearly something different is happening between Sign in the guest and Sign in the account, as illustrated by Sign in to the guest and Sign into the guest both being nonsensical.

sign in, sign into, and sign in to can be transitive or intransitive. Now look at He signed the woman in to his account. Here, woman is the direct object.

He signed in the woman, into his account. Or conversely, The woman was signed in into his account. The latter meets your objective, though by a rather tortuous approach.

(Consider He invited in the women, into his home, He invited the women in, into his home. invite in is separable verb phrase)

In summary, signed in, into .... works in active voice and indicates emphasis or or elaboration. Note the comma. In passive voice signed in into ... is acceptable.