Word for simultaneously smiling at something heartwarming and exhaling air involuntarily

There are many synonyms for sigh but none seem to capture the appreciative nuance that you seek. In some contexts you may be able to use purr.

= (of a cat) to make a soft, low, continuous sound

Cambridge

= a low vibratory murmur typical of an apparently contented or pleased cat

Merriam Webster

The duration of the sound is not defined. Although having feline origins, the word has sometimes been extended to describe a soft appreciative sound:

I could almost hear Julie’s purr of appreciation at my “supplementary”: “As St Augustine reminded us: ‘Lord, make me pure, but just not yet’.”

Catholic Herald

It is just that he is never more vulnerable than when he has just provoked a purr of appreciation

Daily Mail