Many people do Sharkfin Tower in a day. Especially if you car camp at the trailhead and get an early start. In decent shape, you should be able to do:
3-4 hours trailhead to gully (exit from glacier)
1 hour up gully to base of climb
1-2 hours on climb
1-2 hours descending climb
1 hour descending gully
3-4 hours glacier to trailhead
--------
10-14 hours total
So given current daylight length, if you left the trailhead at 5:00am and did my worst-case times, you'd have almost two hours of daylight in reserve.
If you do run out of time, I consider getting to low camp (5600') by dark to be the key to a successful exit. I wouldn't want to find my way down the slabs and to the trail in the dark. The trail from there on out is easy with a headlamp.
Permits can be very hard to get, especially on weekends. We were at the RS at 7:00am last Fri to get a Boston Glacier permit and we were told all of the Boston Basin permits had been gone since 9:00am the previous day.
Steel yourself to listen to the feces disposal speech.