B&W 603S3 crossover

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2.5khz would definitely take things into the right direction and I find 4khz a rather high crossover frequency but I`m unsure that tweeter can even handle anything lower than where its at.

Your tweeter is a 1" dome. It can easily do 2.5kHz.

The designer crossed at 4kHz because it's the easiest to get a nice flat response. Cross lower and you will see that the tweeter response is pretty ugly. The cause of that is diffraction off the front baffle as well as at the baffle edge.

Your best bet is to lower the crossover to 3kHz. But that's not enough. The trick is to shape the tweeter for a gentler roll-off like what Mrcloc suggested.

Your B&W will sound a lot "livelier" with these changes. They won't be as flat as the publish response but at least you'll enjoy your music.
 
Can that be easily accomplished?

Yes, with a bit of design. There are methods, but it is effectively increasing the order of the filter. I have no idea what the crossover looks like, so I can't comment on how to achieve this, but the considerations include what many have said here, for example, baffle influence, phase, etc.

I also agree with what Michael Chua says - to decrease the order the tweeter high-pass filter. That might be a good way of keeping phase aligned.
 
If the design has tried to reduce diffraction issues by running the woofer high, so that the sensitive 4kHz region is covered by a larger driver that is directive enough to steer away from the baffle, it might be best to round the cabinet edges before you cross lower,
 
So a 4khz point was chosen for the sake of minimizing baffle effects on the tweeter. OK, so we drop the tweeter down to 2.5khz.
Take a look at the Low and Low Mid crossover.
What capacitor values will achieve a 2.5k tweeter crossover point and how can you steepen the Low Mid top roll off ? What would the choke value be?
 

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Actually high crossover points are a B&W trademark and have nothing to do with ease of cover design. They do it from their entry level range, all the way up to their flagship.

You could use these measurements as a starting point, but as you can see, the tweeter is quite capable, smooth and extended. As is the woofer. You could manage 1.8kHz with 4th order acoustic slopes or 2kHz if you wanted to be a touch more conservative.
 
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