You're right, the weather forecast does not look good, especially for seeing the northern lights.
But conditions can change quickly here, I don't trust the forecast for more than 2-3 days.
Sometimes the weather is better further east, off the coast, especially when the snow and winds come from south and west.
An option is driving further east to the Finnish border, or even go further into Finland. The sky is often clearer there.
Welcome to Tromso, I wish you luck here!