cat fights

I was sleeping in this morning, recuperating from a weekend of eight Ultimate games at Sectionals, when I was awakened suddenly by loud hissing and growling coming from somewhere in my house. Now, we have an indoor/outdoor cat named Banjo who is, sadly, the cat who gets beat up on the block. This morning I assumed that he was, as usual, sitting just inside his cat door growling at some would-be attacker in the front yard. I was wrong.

As I walked down the hallway, I inadvertently chased an invading black cat into the living room, where he hid under the couch. I went after him and he sped into the kitchen, jumping onto the kitchen window screen. He tore a hole in the screen before his weight pushed it out of its sockets and onto the tomato plants outside. He took off down the street.

And now we have a dilemma. If the neighborhood cats are learning how to get into the house through Banjo's cat door, how do we continue to
a) keep Banjo safe in his own house
b) let Banjo keep his independence and outdoor cat box
c) keep unwanted cats from invading our house
and still have the cat door? Or do we try to re-house train our outdoor-loving kitty? Banjo doesn't like staying inside, but we do not like paying vet bills when he gets beat up. What to do, what to do?