In surface sterilization procedure, you will soak the explant in low amount(0.25%) of ascorbic acid for 5 minutes and inoculate to medium containing activated charcoal or PVP
Try using sterile filter paper disc small enough to hold the callus piece under the callus over the media and remove after few days, if the phenolic exudation/ browning happens only immediately after subculture. What about PVP (Polyvinyl pyrrolidone)? Sorry for the late response but anyway I thought this might be useful.
If it is due to phenolic or any volatile constituent you can use PVP or activated charcoal in you culture medium or even antioxidants like ascorbic acid...if it does not have any volatile content then it is due to leaching you should try re-cutting the cut ends of the ex-plant.
Go for antioxidants, I recommend a combination of 0.5% ascorbic acid and 0.5% citric acid to be added to your media. Also, if available, coconut milk works.
I had similar problem with meristem culture of strawberry. i had followed lots of discussion in research gate and tried different things. finally, i won the battle against browning and fungal problem by simply increasing time in running water (about 2 hours) after treating with sodium hypocloride along with Tween 20. i used to do very frequent sub culture , finally i don't need to spend time like that. it really worth spending two hours in explant preparation than wasting media afterwards.