What is the best way to tame a slightly bright speaker?

Get speakers with non-metallic cones.

I would strongly advise against any kind of messing around with crossover values. Any speaker worth its salt (certainly something as high-end as the TAD) has a network precisely tuned for attenuation, corner frequency and phase alignment in the context of the speaker’s impedance. Changing any of those values - caps, resistors, inductors, sometimes even minute changes in resistance from different hookup wire gauge - will throw off that tuning and have unintended side effects. You can’t just "pad" a driver in a typical crossover-based system without shifting other parameters.

The room treatment and positioning suggestions are good, and based on measurements from the Stereophile review you might be able to use the directionality of the coax driver to your advantage. But I think speakers have an intrinsic tonality that your ear will always pick up on regardless of the particulars of the setup. I can hear this from speakers at shows - even if the room is totally dead or bright or unknown, you can still kind of tell what kind of energy the speaker is putting out. In the case of the TADs, in my run-ins with them I noticed a bit of hardness/ringing to the upper midrange/lower treble that I found fatiguing. I wonder if this is what you might be picking up on rather than an actually elevated treble response, especially given that the measurements from the aforementioned review seem quite flat in the upper frequencies, and if anything slightly tilted down in the top octave. If that’s the case, ameliorating with a different amp and/or cables may be more effective. I hesitate to go down the cable rabbit hole in a forum like this, but I’d avoid anything silver or tinned. Audience is a safe bet for natural, un-fatiguing highs, and I’ve also been impressed with Crystal Cable in a few different setups.

Perhaps I missed it in all of the chatter.  What are you running for a preamp, DAC, turntable, etc.  Seems like a lot of very complicated solutions for a simple task.

This may be redundant but get a multi band (31?) equalizer. First line of defense for room treatment especially if your room is small. The judicious application of equalizing bright speakers will more than compensate for the addition of yet another electronic component in your chain.

Also makes a great notch filter for bass traps in a small room as well. I use one and find it indispensable. Not the "purist's" choice but it works and it is inexpensive. Have fun.

I have similar challenges and posted similar questions, got similar responses, here's what I have and have not tried so far - 
1) Modify the crossovers - probably a good route for those with sound understanding of circuit design and comfortable working on gear, but that's not me, not yet anyway.
2) Change amps - I went to a tube amp and this helped, made a marvelous improvement in the mids (no surprise) but high end smear still there.
3) Room treatments - a never ending pursuit.  Adding heavy draperies on the front wall, furniture in the corners, diffusers and/or absorbers at first reflection points on sidewalls have all helped.  The draperies made a huge difference in my room, but I have a window on the front wall. 
4) Speaker placement - I used the Cardas approach, then experimented with angle, ended up about 5 degree toe-in.  Some report much improvement by moving the speakers to the long wall, but in my room that didn't help and made it nearly unlivable, but my room is only 11.5 feet wide.
5) Change/improve source.  I have a PS Audio DSD and some suggested going to a "warmer" DAC, Mac or Bryston, but so far I have chose to stick with the PS Audio.
6) Power - upgrade everything you can in the power supply.  For me, Cardas Clear M AC cord for the amp, plugged directly into wall, made a nice improvement.  Listening room is already on it's own circuit, I will be trying a higher grade outlet next.  Fuses maybe?  
7)  Isolation - improving isolation of the amp helped a surprising amount for me, so I will keep tweaking with other components too.
8) Different speakers have been recommended (I have Thiels) - well that's just silly, as I want to optimize the SQ of my current speakers.
9) Room correction components/software - kinda like the speaker crossover modifications, this is just beyond me at this point, I'd rather spend my audio dollars on other things for now.
10) Subwoofers - same as above.  Someday probably, but not in the near future.
11) Cables and interconnects - I have avoided going down this rabbit hole so far.  

Bottom line - there are many ways to skin this cat.  The good news is that most of the things I have tried so far, while not always helping solve the treble problem, have led to other very nice improvements, like detail, resonance, and staging.