The 23-year-old faces six offences, including two counts of encouraging or assisting offences and four counts of taking part in an unnotified public procession.
During a hearing at Belfast's Laganside Courthouse on Saturday, Bryson, who is from Rosepark in Donaghadee, was remanded into custody to reappear via videolink later this month.
District Judge Bernadatte Kelly told Bryson that she shared police concerns that if released he would pose a "flight risk", having heard how he refused to hand himself over to officers knowing they were looking for him.
Bryson was arrested on Thursday by officers from Operation Dulcet, the PSNI taskforce set up to investigate the disorder surrounding the Union flag protests.
Former BNP fundraiser, Jim Dowson, also appeared in court. The 48-year-old from the Burn Road in Comber is charged with one count of encouraging or assisting offences and five counts of taking part in an unnotified procession.
He was released on bail on condition he surrender all communication devices, including SIM cards and USB fobs.
Judge Kelly said Dowson was not to use a telephone except in an emergency situation, added that if anyone communicated on his behalf, Dowson would be remanded into jail.
He has also been banned from entering Belfast or attending any procession.