This was too good not to dig my teeth into!
Okay some clarifying vocabulary here:
"任務" can mean mission, duty, role or task
"捨てる" can mean to discard, throw away, or cast away (and only "sacrifice" like you sacrifice a chess/go piece)
"自分" refers to the "self", one's self, sense of self or identity
So there's a lot of fun stuff going on in this and Endo is a genius. Here goes:
First line isn't that interesting. It just states "for the sake of my mission my safety is [sentence cuts off, but implies 'most important']"
Buuuut next sentence has rhythm with this but changes so:
The fact Loid uses a word for "mission" that can also mean "duty" [IE is it duty to his mission, his country or his family??] is underlined by Endo in the first section of this where "for the sake of my mission" and "for the sake of my family" criss-cross - so the word for "mission" is implied to mean "duty [to my family]".
So second line is closer to "For the sake of my family I have to throw my self away, but"
See what I mean about double meanings being important?
There's an implication here that the self that is twilight has to be discarded for the sake of Loid's family.
No matter what OP says, however, there is a phrase that puts "family" into an object construction using the particle を. But it is ridiculously cut up and like the former sentence ends on a "but". It goes
"For the sake of my mission - I must throw away - my family - but"
And then it ends on four lines that just goes "for the sake of my duty/mission". Which, again, can mean both his duty to his family as Loid or his duty to his mission as Twilight"
Buuuuut the second to last one has the first kanji in "mission" cut out, and the kanji that remains "務" means, more clear cut, "employment" or "to fill a [work] position". So the second to last sentence implies
"for the sake of remaining as a spy".
Before cutting back to "for the sake of my duty"
So it feels like there's a massive internal battle with the [to put it frankly] brainwashing Twilight went through during his spy training going on here. Much more than even the English language translation could smoothly convey.
IT feels like the "Self" that is Twilight is being consumed by the "self" that is Loid, and that Loid who loves his family is becoming a real person. And I find that veeeeeery interesting