Yes, loss scaling is a valid approach. Basically, cross-entropy formula matches categorical distribution density (NLL), and MSE - gaussian with scale 1. So, rescaling loss corresponds to adding gaussian scale as either (trainable scalar) parameter or hyperparameter. Similar logic applies to other loss functions, if you can imagine them describing log-densities of some probability distributions.
OTOH, such networks are multi-task ones, with parameter sharing. So it is possible that tasks will “compete” for shared parameters. This is not always a problem, but when it is, loss scaling won’t be a panacea…