There is a guage that measures how soft lead is. I prefer softer lead.
Antimony is added to make lead harder among other things but in certain barrels it can lead to flyers as the barrel gets a skin after each shot that is then picked up by another pellet somewhere down the line. It tends to be a pedictable pattern as to when a flier appears but not always.

IIRC JSB includes antimony in their production but the % may of course be tweaked depending on various factors.

There is no need to use a phosphur bronze brush on airgun barrel. A brush like that is to deal with burnt powder residues and fouling from copper jacketed bullets. It does nothing apart from wreck probe o ring seals.

Some barrels like to be clean. Some dirty. Do a 5 to 10 shot group and keep it to compare later at a distance that begins to challenge ypur group size but alloes you to be consistant in placement. Too close and you wont see the group change. Too far and your errors disguise the real grouping potential.

Pull trough the barrel with .22 lr patches making sure NOT to touch the crown. The chordd can very easily damage the crown the most vital bit of a barrel.

Usually, a barrel might need to lead up again. This might be 40 shots or so either way. Do another 10 and see if the group has tightened or loosened it will say if the barrel likes to be clean or run in with lead.

Every 100 or 200, do another test string to see if its still tightening, staying the same or loosening again. You will work out how many shots you can get away with and how many shots you can shoot before another clean is necessary.