Right ML mode and metric to minimize FN and FP on imbalanced dataset
If you use class weights, I don't think it would make much of a difference which model you use (regarding the imbalance). If you are familiar with python, I would suggest this sklearn function that could help you out in computing the class weights. However, I have found that oversampling (or undersampling) often works better than class weights. Consider using SMOTE for this.
While there are metrics and loss functions better suited at handling class imbalance (GMS, quadratic kappa, etc.) I don't think you would have an issue with using any loss 
if you considered oversampling your data.