I've played with some bypass caps and I really like the Miflex caps, they add a little extra warmth/richness to the overall sound of the tweeter, but they will need ~400hrs before they settle.
Adding a high-quality polycap as a bypass typically only improves clarity, but there are some really tiny differences between brands/varieties, but it's maybe 1-2% here or there.
film & foil caps, like a Miflex, Duelund, or Jupiter tend to impart some of their tonal characteristics. Copper adds a little warmth/richness, silver adds clarity, but may come off as clinical/grainy. Tin tends to be more neutral, and aluminum is somewhere between tin and silver, depending on the application.
Oil-damped caps often need 400-500Hrs to fully burn in and may be a little "hard" or "flat" sounding at first depending on the cap/brand, but will relax and open up over time, some caps are also rather "dramatic" during burn-in, and can sound great one day and really rough/edgy the next, and the day after, the soundstage may seem flat as cardboard or deep as the ocean.
Also, the larger the bypass cap, the more influence it will have over the main cap.
For a tweeter circuit I generally recommend sticking to 0.1uF, but you can go slightly larger to 0.15uF or 0.22uF if you feel you're wanting more tonal influence, or down to 0.047uF or even 0.01uF if you want less.