As there are only a few accidents and he is mostly dry, I would continue with no pull-ups, just waterproof the mattress and put absorbent bed pad under him. You can buy washable ones or disposable ones.
Regarding the tantrums, I guess he's just being hard on himself. You can explain that you aren't cross, just that sometimes he will have accidents. You could look back to the day and its events when he is wet that night, to see if there is anything that could have been the cause i.e very tired, starting a growing spurt, upset about something, drinking more than usual before bed, been to a party and had something fizzy or a drink more likely to make him wet like blackcurrant, starting an infection.
It helps if you get DS to drink lots during the day to hydrate himself well and train his bladder to stretch, then reduce drinks after teatime and then just plain water. Make sure he does a wee before going to sleep, not just before bed, as there may be a long time between especially if he is having a story. So toilet before getting into bed, story, toilet again. I think he's so nearly there, it would be a shame to go back to pull-ups.