I have ridden to Logan Pass three times. Definitely try to do it.
Whitefish up to the top of Logan Pass and back to Whitefish in one day would be tough. Even if you were to take U.S. 2 (there is a narrow, shoulderless section which makes a different route advisable), it's 27 miles from Whitefish to West Galcier. From there, its around 1-2 miles to the start of GTS near Apgar. And IIRC, from that point it's 21 miles to the summit. So you are talking 100 miles round trip.
As noted, there are restrictions on when you can ride your bike on GTS Road. Check the park's official web site for details. Also, the best time to ride it is early in the morning, before the traffic picks up.
If your last day to Whitefish is a short one, I would continue on to the park and stay at either Sprague Creek or Avalanche Campground. Both have hiker-biker spots for around $5/night. Otherwise stay in Whitefish and ride to the park the next day.
Ride up to the pass early in the morning. (You have to make it up by 11 a.m.) Try to do one of the short hikes like the one to Hidden Lake that starts from behind the Logan Pass Visitor Center. Then come back down. You can then either stay another night in the park or ride back to Whitefish the same day. Note that, because of the bike restrictions, you will have to wait until 4 p.m. to exit the park on your own. (Coming down the west side I think you can ride as far as Sprague Creek before 4 pm. Again, check the park's site as it's been two ears since I last rode it.) As noted, there is a shuttle.
Finally, flight schedules may dictate your timing. When we flew out of FCA, most the flights were in the morning.