There is no difference in the log prob between LSTM and transformers. It just depends on the number of updates you make (batch size, epochs), if you train it to overfit, then your log probs will be pushed closer to 1 or 0.

But after training you can recalibrate the temperature of the softmax on the test set and still get meaningful confidence scores (temperature calibration). Or you can use a variation of cross entropy called Focal Loss that will leave your logits un-squashed.