B&W 802D1 - How can I tame the brightness?

The best way to fix speakers that are not pleasing to you is to get new speakers. There are so many good ones to choose from that it can be overwhelming. You might  start by auditioning the Sonos Faber Serafinos 

These are excellent speakers with a touch of warmth that will never be fatiguing. Some people might find them “soft” while others will think the presentation is exactly what they want. No substitute for listening.
 

"I can tell by your post you've never had a pair of these speakers."

LOL!  As a matter of fact I do own a pair of these speakers.  I know them intimately and know their strengths and weaknesses.  With these, or for that matter any speakers, "fooling with the room" is everything.  You have admitted your own limitations in your post.  Make no mistake, though, the OP has a room problem, and 802 D diamonds when properly set up are not overly bright or harsh.  

Unfortunately, that brightness is due to the tweeter and to my ears is very tiring. 

Onehorsepony, at 8:26am you posted ’set up could be an issue’. Yet you seem to imply that it is a problem you don’t care to deal with.

There are several people on this site who could help you with the entire set up issue if you choose not to do some independent research yourself. To get this input you need to describe, with specificity, your room dimensions, placement of windows and other openings, and wall furnishings, ceiling, and floor coverings. You then need to identify the present location of your speakers and listening chair in inches so folks can tell where they are and how they are canted. This information could start a meaningful dialogue which might solve your problems, or at least, minimize them.

Or in the alternative you can take the advice of posters who own, owned in the past, or make a living off selling what they recommend, without any of them knowing what causes your issue other than your generic description, or you knowing as much about their system as they do about yours. Your choice. 

Before you sell them at a loss, try facing them straight ahead (no toe-in).

I did that with my 803 D3 and was very happy with the sound.

It might work for you too.