You need an appropriate size overcurrent device for the feeder conductors at the SUPPLY side. The "main" breaker at the subpanel is entirely spurious as far as the code goes.

"... All I need is to tap into two phases of the buss bar with two lugs at the end."

You can't legally do that. And you probably can't mount a 200A breaker on those busses. Look on the panel's label for stab rating. That's the maximum size breaker that fits in two spaces. Double it for a four space breaker.

I need to know if the garage is attached to the house or if it's a seperate structure.

And on the left/meter side of the panel. Are there more lugs on the metered busses? Not the service side busses.

The panel takes QOB breakers (other than the main) and you can get those up to at least 150A for that panel. There are no "four space" 240 QOB.

There's no listed mod to add breakers other than to regular buses.

Looks like I cannot upload more than one image, so I will post several messages with images. This image shows the actual panel. It is outside the garage wall. Garage is attached to the house. The subpanel in the garage will be on the opposite side of this panel. As you can see, there is already a 200 Amp main breaker driving the buss. There are no more open lugs on metered busses on the left side.

One option can be to remove the buss bar and just use the main breaker as a disconnect that is directly feeding the garage subpanel. Is this something acceptable? There are 6 screws keeping the buss in place. It can be easily removed without affecting any other component. Of course, it would be better if I can keep the buss in place.

By the way, I learned that what I need is called feed-through lugs, not sub-feed lugs.

Attaching another picture

And one more picture...

Great pictures.

"... One option can be to remove the buss bar and just use the main breaker as a disconnect that is directly feeding the garage subpanel. Is this something acceptable?"

Your inspector has the final say. IMHO it is acceptable.

"...Of course, it would be better if I can keep the buss in place."

You could check to see if there is a double lug kit for the top of the breaker busses. Double lugs are what the four wires are coming out of on the left. Replace the single lug with the double lug. Then, for each double lug, the jumper from the 200A breaker would go in one side and the feed to the garage in the other. It acts just like a feed through lug except all the action happens at the top. Always torque nuts and connectors to the proper specs.

The IT-250 I suggest at Mike Holt is just another way to make the same connection.

Great! Thanks glennsparky.

You are doing this all to avoid a 2nd 200A breaker? Just run to the 2nd panel and have both breakers. It will work just fine, right?

Again, you MUST have a breaker on the supply side of the feeder here. None of the dozen or so exceptions applies here.