Brushing without toothpaste will not IMO make lips heal any faster. I tried it. You stand less chance to physically interfere with the scabs which is a good thing. The white line build up is just saliva and the lips don't feel dry because the cleaning chemicals in most toothpastes will do that. It's hard not to use toothpaste for weeks/months though. It's not an allergy from the toothpaste, I've trialled this and can confirm this, it's just some are more abrasive to the scabs than others.
Eliminating Sugar may make a difference to the saliva produced in the mouth but has nothing to do with the EC. Daniel miller used toothpaste throughout.
Trial the probiotics if it means you can tick it off.