From doing research recently it seems that some of the guys who enlisted/joined the Royal Marines this was actually a preferred method to transportation by enforced service. I don't know if that is the case here. William was Irish and his brother Michael Byrne was a convict sent here to NSW in 1799 after being tried in Kildare in 1797 (coincidence??) not sure
"Mass recruitment in Munster and Leinster from the spring of 1797 then broadened the base of the unified movement ... By the eve of the Rebellion ... a paramilitary force boasting a nominal strength of 280,000 members. ... Leinster United Irishmen recruited in the big drive of 1797 ..." ("Desperate and Diabolical - Defenders and United Irishmen in Early New South Wales" by Ruan O'Donnell)
Jails of assize Home Circuit April 1797 were reputedly "filled with men charged with Defenderism ... Farrell Cuffe has been found guilty ..." (Saunders Newsletter 26th April 1797). Farrell Cuffe was a teacher in King's County (County Offaly) who was convicted of administering illegal oaths.
The Insurrection Act, introduced in 1796, remained in force for several years. It gave magistrates sweeping powers of search and arrest.
Were the Byrne brothers R.C.? Catholics were allowed to join the British Army from late 18thC.