Yes, going to Alaska is a bit expensive but after your trip you'll be amazed by Alaska the place more than just a destination bike ride.No need to worry about over growth on Resurrection Trail as most of it is above tree line or on well maintained trail. Can easily be ridden in a full day  but make it a bit of adventure a get the Devils Pass or other cabins on the trail and stay out there overnight. Johnson Pass and Russian Lakes trail are more likely to get grown over Primrose and Lost lake not as much. 

If your willing to to look at other biking options I can say there is no better ride than the 90 miles of gravel through Denali National Park from Wonder Lake / Kantishna to the Main entrance. It's all road, not flow single track but its pure Alaska wilderness. Can be done in 2-3 days if your looking for more of that continue on the Denali Highway for 135 miles from Cantrell to Paxson.

The must ride sections are Resurrection Trail, Johnson Trail and Upper Russian Lake trail. Deveils pass is a nice add on wing to the Resurrection Trail but the other trail while less know and not as biking friendly still worthy in their own ways.

Re the Bears: Russian Lakes trail is a bear highway with a long history of bear people encounters with considerably more bear activity while salmon are running. I would avoid or be very wary on that trail in late June and all of July. All other sections, take normal bear country precautions i.e. bear spray, make noise etc. In over 20 trips on these trails (except Russian River) where I've seen much bear scat but never a bear close enough to get concerned about.

The Alaska Railroad offers a option for getting t/f Anchorage-Seward it not exactly fast ~!4hr but a very nice ride and the are willing to take bikes in the baggage cars.

Consider riding to hope if you have some time or ride from Anchorage to Girdwood and then Girdwood to Hope. Both are fun warm up rides mostly paved trail from Anchorage to Girdwood. Nice why should from Girwood to Hope cut off and not much traffic on the Hope cut off.

Some trail info

https://www.mtbproject.com/directory/8006867/southcentral-alaska