Word for being "determined to do something despite knowing that you are doomed"
I think resolute (adj) or resolve (noun) could be good options.
resolute [adj.]- firm in purpose or belief; steadfast
resolve [noun]- fixity of purpose; a strong determination to do something
In your sample sentences:
The captain's speech filled him with resolve.
Despite knowing that he would be dead before the day was out, the resolute man marched on. (or maybe "...the man marched resolutely on.")
I think it captures some of the grit/tenacity from your Sisu example, with a bit of an aspect of stoic resignation/"grave determination", as you said.
I believe you are describing courage.
The ability to do something that frightens one; bravery:
'she called on all her courage to face the ordeal'
Applied to your examples, you get the following:
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"The captain's speech filled him with courage."
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"Despite knowing that he would be dead before the day was out, the courageous man marched on."
How about resignation and resigned? They work well in your examples:
"The captain's speech filled him with resignation."
"Despite knowing that he would be dead before the day was out, the resigned man marched on."
From Dictionary.com:
resignation: an accepting, unresisting attitude, state, etc.; submission; acquiescence: to meet one's fate with resignation.
resigned: characterized by or indicative of resignation.