I suppose a determinist might, first of all, acknowledge that he can’t save anyone from “hell”, since that is God’s job, via Christ crucified. Christ is the Saviour of the world, not man. To Him be ALL the glory, not half & half with humans, imaginary human freewill or idolators. OTOH if freewillers feel it is their duty to save themselves & save others, that’ may be putting heavy burdens on people, likely accompanied by many fears as well.
Re those “predestined to Hell”, or rather corrective chastening or punishment, nobody knows who they are & even if we (freewillers or determinists) want them saved from such, God clearly knows better, and in any case there is nothing either of us could do about it.
As for “hell” itself, i suppose it may be much worse in the freewill than determinist view of universalism. To some freewillists it may take many ages of tortures before the wicked finally choose God of their own freewill. The deterministic God has no such problem as freewill to deal with.
Additionally, the free-willy God has a problem keeping all saved, if He can ever get all saved [which is certainly questionable]. For if they still have freewill, is He going to take freewill away from them to keep them saved, or is He going to let them still have freewill to choose to be lost?
Per the determinists’ motivation to preach the good news of the gospel - from the merely human, or relative perspective - it is because
of some of the common usual reasons one hears, that it is commanded, it is the will of God, because one is called for this purpose, because you love Christ & love to proclaim His glad tidings of great joy, which shall be to all people.
OTOH there is the divine and absolute perspective, which is the love of Christ constrains us & works in us to will & to do of His good pleasure (Phil.2:13). One being chosen by God to this ministry is, like our salvation, not of ourselves, but a gift of God, not of works, lest any man should boast (Eph.2:8-10). For it is not of the will of the flesh, nor of the WILL OF MAN, but of God (Jn.1:12-13).