Can the fixed shares be varied by agreement?

Agreed. The four schools hold that the shares are not the testator’s to move. A bequest to a person who already inherits does not take effect — “Allah has given every rightholder his right, so there is no bequest to an heir”, from the Farewell Pilgrimage sermon — and Ibn Rushd records the schools as agreed both on that and on the consequence, that a bequest to an heir cannot stand where the co-heirs refuse it. A will therefore cannot be used to disinherit a son, to top up a daughter, or to attach conditions to what a fixed heir takes. What the shares can be varied by is agreement among the heirs after death. Once the estate vests, each heir owns their share outright and may give it, sell it or trade it, so adult competent heirs may settle among themselves — one takes the house and compensates the others. Ibn Qudama names Ahmad, al-Shafi’i, Abu Hanifa and his companions, and Malik as all giving effect to the heirs’ consent once the testator has died. The dissent Ibn Rushd records against that — the Zahiris and al-Muzani — belongs to none of the four. The distinction that carries all the weight is whose decision it is: the testator may not redraw the shares, and the heirs may only give away what has already become theirs. A minor’s share cannot be settled away by anyone. We checked this point; the schools do not differ on it.

Shared evidence: Sunan Abi Dawud 2870 — Abu Umamah, from the Farewell Pilgrimage sermon: “no bequest to an heir” · Jami‘ at-Tirmidhi 2120 — the same report; al-Tirmidhi grades it hasan sahih · Surah an-Nisa 4:11 — “an obligation from Allah”.

Not covered by this finding: what happens when a court varies the shares rather than the testator — a British Columbia wills-variation order under s.60 WESA, or an Ontario dependant support order — which is a contemporary question the classical texts do not reach — not yet researched school by school. Treat that as open and ask a scholar of your school.