We have let them be friends means the same thing as we have allowed them to be friends
I agree. It is less formal, but would be the most likely way of expressing it.
Why is 'become' wrong?
That would be "Vi har tilladt dem at blive venner."
Ikke at de behøver vores godkendelse til det, men stadig.
heh... typical of the Danish... they don't make friends with anyone... they have to carefully choose them.