¿Por qué esta consulta no funciona en SQLite pero funciona bien en MySQL y MSAccess?

En SQLite, obtienes una columna de incremento automático usando INTEGER PRIMARY KEY .(Y si confía en los valores reales de payers.id en la consulta, debe proporcionarlas explícitamente).

No debe intentar anidar uniones cuando no es necesario:

SELECT payer_payment.payer_id,
       Sum(payer_payment.amount)                             AS total_paid,
       Sum(payer_payment.pays * payments_share.single_share) AS fair_share
FROM       payers
INNER JOIN payer_payment
        ON payers.id = payer_payment.payer_id
INNER JOIN (SELECT payment_id,
                   Sum(amount) / Sum(pays) AS single_share
            FROM   payer_payment
            GROUP  BY payment_id) AS payments_share
        ON payer_payment.payment_id = payments_share.payment_id
WHERE payers.user_id = 1
GROUP BY payer_payment.payer_id;

SQLFiddle