A word for "remorse" but without guilt

Solution 1:

The closest that may come to the idea of a feeling of sadness without guilt, ("Well, I don't like it, it's bad, but it could not be helped") would be
regret [WP]

Regret is a negative conscious and emotional reaction to personal past acts and behaviors. Regret is often expressed by the term "sorry." Regret is often a feeling of sadness, shame, embarrassment, depression, annoyance, or guilt, after one acts in a manner and later wishes not to have done so. Regret is distinct from guilt, which is a deeply emotional form of regret — one which may be difficult to comprehend in an objective or conceptual way. In this regard, the concept of regret is subordinate to guilt in terms of its emotional intensity. [emphasis added]

Solution 2:

The mood might be described as somber and the person might be resolved to their task

From Merriam Webster:

somber : 2 a of a serious mien : grave < somber dignitaries >

resolved : to make a definite and serious decision to do something