How do you finish off a woven bracelet so the clasp doesn't look clunky?

A tapered end section is honestly the biggest fix for the bulk problem. I weave the last quarter inch or so down to fewer beads per row, so on a band that's mostly eight or ten beads wide I'll drop down in steps, then land on just two or three beads where the clasp actually attaches. A lobster claw sitting on a two-bead tab looks like part of the piece instead of a hunk of metal bolted on. If I'm doing a flower or diamond pattern I'll sometimes repeat one small motif from the body of the bracelet in that narrow section, which makes the whole end read as deliberate instead of an afterthought. The flopping to one side issue on your chevron piece is usually about where the tension sits, not the tension itself. If the clasp anchor point is even slightly off center, the band twists on the wrist. I stitch through the center bead or the middle of the last row and pull that thread path snug before I even bring the clasp into play. Also worth checking that your end row is truly level, because one wonky bead will torque the whole thing.

On crimps and accent beads, that works fine for strung pieces but it never looks right on a woven band, the stiffness is too obvious. Bead caps help a bit, but what helped me most was switching to a folded over end loop instead of a straight tab. You weave back into the band a couple rows, fold the tab under, and stitch it down, so the clasp hangs off a little reinforced loop that flexes with the wrist instead of poking straight out.

Extender chains, I only add them if the bracelet needs the adjustability. On something delicate like your flower piece a chain can look fine if you put a couple of seed beads or a tiny drop bead on the free end so it doesn't read as bare hardware. On a wider band they dangle and snag on sleeves, so I skip them and just size it properly or make the clasp side slightly longer with extra rows instead. One more small thing, use a finer thread on the finishing passes than you wove the body with. Bulk at the end is often thread buildup as much as hardware.

Before fancy tab finishes were everywhere, we just swapped the clasp itself. A little button or a single fancy bead with a loop of delicas on the other end blends right into beadwoven bands, no hardware sticking out at all. I picked that up back when most of us were working off graph paper patterns and it still looks cleaner than any claw on a dainty piece. Only catch is you need a decent sized opening, so test the fit on your wrist before you stitch the loop down, not after like I did on my first attempt.

The folded loop idea makes sense to me but I worry about strength. I once tried making my own little loop out of beads on the end of a bracelet and the thread sawed thru after a week of wear, the clasp just came off in my hand. Is there a limit to how many times you can pass thread through those end beads before it stops holding and just starts bulking up again? Also does the loop thing work if the clasp is on the heavy side, or does it pull the band out of shape over time? My stuff keeps falling apart right at the end where all the stitching piles up.

I mostly string things so take this with a grain of salt, but I've been wondering about magnetic clasps for woven pieces too. On my strung necklaces they sit nice and flat, but I'm not sure how they'd hold up on something that flexes as much as a woven band. Do they pop open when the bracelet bends, or are they solid enough for daily wear? Also not sure how you'd even attach one without it looking chunky, which seems to be the whole problem here. Might be a dumb question but this seemed like the right thread to ask.

Update: tried the tapered end like the first reply suggested, dropped down a couple rows at a time and the clasp finally sits flat. Huge difference. Now I'm eyeing the button idea too but got stuck on the loop side. How do you reinforce it so the thread doesn't saw through like that happened to one poster here? Fireline, extra passes, or a dab of glue under the last beads?