I have no idea which specific losses you are using. Anyway, the following may help solving your problem.
Just define your loss_total as:
loss_total = alpha*loss_1 + (1-alpha)*loss_2

and finetune the hyperparameter alpha in the interval (0,1).
If you say that loss_1 dominates the total_loss, then you’ll have o choose an alpha that is closer to 0 than to 1. Maybe you can start with alpha = 0.4 and, if it still does not work, decrease it by a fixed step of 0.05 or so, until you get reasonable results.
However, before doing this cross-validation, you may wish to try training with alpha = 0, just to make sure that you can in fact minimize loss_2. There could be a bug in the definition of loss_2 and this procedure allows you to check that.