whether we need the visa to give notice of the marriage
Yes. You cannot give notice for a marriage or civil partnership without a Marriage Visitor Visa.
(Under a Standard Visitor Visa) You cannot:
...
marry or register a civil partnership, or give notice of marriage or civil partnership - you’ll need to apply for a Marriage Visitor visa
(While your fiancé does not need a Standard Visitor Visa under normal circumstances, the list of permitted activities is the same for visa-exempt visitors.)
come back on the same visa to get married after
Yes, if the MVV is not single entry and both entries are within the validity of the MVV. As an entry clearance or certificate, which has the force of a leave to enter, the MVV authorizes both giving notice and the actual marriage within its validity on UK soil.
It is unlikely for the visa to be single entry for a non-visa national unless there are other significant concerns, but you may want to provide concrete plans and reasoning to explicit ask for a multiple entry visa.
The visa is normally valid for six months, so you may want to apply for it not too early (but also not too late, unless you are prepared to pay the priority processing fee) and clearly outline your plans.