I did read the entire letter. But my comprehension of the phrase "that measures, otherwise unconstitutional, might become lawful" means that Lincoln understood them to be illegal for him to perform. Or why else express the sentiment that they "might" be legalized (ex post facto as well, still unconstitutional).
I think you are stumbling over an archaic (i.e., precise!) use of "might." It is the past tense version of the following: "Measures, otherwise unconstitutional, MAY become lawful, by becoming indispensable to the preservation of the constitution, through the preservation of the nation." And I think the meaning of "may" in such a sentence ("might" in the past tense) means what "may" means when we tell children to use "may" instead of "can" -- as in "Can I have dessert?" being corrected to "May I have dessert." "May" here means something like "has the power of legitimacy, not of raw ability."
Obviously one can disagree with Lincoln's judgment, as he himself acknowledges in the immediately next sentence -- "Right or wrong, I assumed this ground, and now avow it." But I don't think that there is any suggestion in these sentences that he thinks what he did was unconstitutional or unlawful when he did it. Rather, he is saying that he believes that the circumstances he describes really do have the legitimate power to convert a presidential act from unconstitutional and unlawful (in other, perhaps normal, circumstances) to being genuinely constitutional and lawful at the time it is taken. Am I being clear?