i don t think that any of the songs without words can be the piece you re looking for, they re pretty much static, most times the same way they start, they end (i mean, they tend to miss those "dramatic sections" you were talking about...anyway check this link in his channel you can find all the songs without words with sheets to follow what if your granny is misremembering the composer? cause even if i don t have that deep knowledge of mendelssohn' music all i could vaguely think about with such a description was 'Rondo' Capriccioso', but i don t think it is that piece because it s quite demanding...what about Ravel ' Oiseaux Triste' ?
I know this is VERY far fetched (especially as the name is completely opposite to what you suggested) but your description of birds and sun bursts really remind me of Grieg's Nocturne.