No lord ram is not always forgiving . He do give chance anyone before reaching any decision . When he see anyone crossing limit inspire of giving chances then he give punishment
Ram principles was first try to make understand calmly the person who is wrong
Here are many incidents where Lord ram did not take any decision without thought and hurry and even did not take decision in extreme anger
Like
When he face taadka he did not kill her for first time but he gave many chances to tadka so that tadka stop attacking but tadka continue attacking so ultimately taadka had to take hard decision
2 he give time 3 days to ocean to ask way to cross ocean but when ocean did not listen to his request then he decide to dry ocean with one arrow
3 he gave chances so many times to raavan too
Here ram first priority to deal anything calmly instead of just attacking . And I think it is quite sensible approach
As concern to laxman he set great example how should no one tolerate in justice at any cost
Here Lord ram also not tolerate justice but here Lord ram first prefer to give chance but laxman don't believe in giving chance most of time and I think principles of both brother are harmless