Russell,
Thank you for your post. Interesting question, but one I try to avoid for a couple of reasons. The first is that I'm not an engineer, and thus am vulnerable if I give an opinion on something so divisive, and second, this divisiveness extends across the whole spectrum and can produce enormous friction and name calling.
Have a look at the 'Audibility of Output Coils' thread in the SS section of DIYaudio.com to get a handle on just how acrimonious these things can be...... it ain't pretty.....

Capacitors color the sound. The best of them, in my experience, have been the teflons and the polystyrenes, which are difficult to detect.
OTOH, some caps color the sound objectionably, like some metallised polyester caps, and some color it euphonically, like the Jensen paper in oils and the RTX hybrids, ps and pp foil, and the Sprague orange drops widely used in guitar amplifiers. It is thus OK to say 'depends on the cap'.
Musical instruments play to a certain tone, and the electronics and source instrument are carefully tailored to that tone. A Fender Stratocaster and Tweed amp has a certain sound, so does a Gibson with a Vox, which uses EL84s in Class A. Traditionally, and even this has caveats with tube amps, the hifi amp strives to be faithful to the source, so we want our caps to be without color. Yet there is evidence even in hifi, and a good example is my AKSA which is slightly colored, that people like a bit of color there too.
The argument is vehemently put in objective engineering circles that there should be absolutely no measured distortion in a high fidelity amplifier. Yet even in direct coupled designs, there will be a blocking cap, and this will add color, not to mention the many caps in the passive crossovers of almost every commercial speaker in the marketplace. As a general rule, less busy music like jazz quartets and simple vocals, sound very good on somewhat distortive systems, such as single ended tube amps. But busy music, like large orchestral pieces and rock concerts sound better on very low distortion solid state amps. And so, while the focus is correct and often ideologically driven the reality is something different. And besides, people's tastes do vary enormously.
And so, my answer is not definitive. Objectively, the cap in series with the tweeter will increase distortion, but if the cap is well chosen, the distortion will be limited to H2 and a smidgin of H3, and may sound very nice indeed. Subjectively, the issue is clouded by psycho-acoustic preferences.
Simply put, whatever floats your boat!!
As you can see, I'm an artful dodger. I'm sorry about this, but no one answer is strictly valid.
Cheers,
Hugh