Her Majesty's Government is a democratically elected executive, to enact the will of the people through Parliament. The withdrawal agreement has been through the due process. Couldn't they present this 'as is' to the EU for signing without getting Parliament's approval? (Getting the parliament to an agreed majority solution will not happen).
This plan almost certainly couldn't work, since section 13(1) of the enacted version of the European Union (Withdrawal) Act 2018 states
13 Parliamentary approval of the outcome of negotiations with the EU
(1) The withdrawal agreement may be ratified only if —
(a) a Minister of the Crown has laid before each House of Parliament—
(i) a statement that political agreement has been reached,
(ii) a copy of the negotiated withdrawal agreement, and
(iii) a copy of the framework for the future relationship,
(b) the negotiated withdrawal agreement and the framework for the future relationship have been approved by a resolution of the House of Commons on a motion moved by a Minister of the Crown,
(c)a motion for the House of Lords to take note of the negotiated withdrawal agreement and the framework for the future relationship has been tabled in the House of Lords by a Minister of the Crown and—
(i) the House of Lords has debated the motion, or (ii) the House of Lords has not concluded a debate on the motion before the end of the period of five Lords sitting days beginning with the first Lords sitting day after the day on which the House of Commons passes the resolution mentioned in paragraph (b), and
(d) an Act of Parliament has been passed which contains provision for the implementation of the withdrawal agreement.
Since it takes statue to override statute, a government will need parliamentary approval, or a new act to modify the rules in order to have an agreement with any legal force within the UK.