You have a couple of choices for a southbound CDT route through Glacier. The first starts in Waterton Lakes National a Park in Canada, crosses the border at Goat Haunt, and follows the Highline trail through Glacier. This route is open at the earliest around Aug 1, as it depends on snow conditions and the Ahern Drift trail segment being blasted out.
The second starts at the Chief Mountain Customs Station. The trailhead is about 50 yards south, and has a large parking lot. When we were there around 7/21 there were about 40 cars there. This route goes to Belly River, then Elisabeth Lake. After that you can go via the Ptarmagan tunnel if it is open, or the longer route via Poia Lake to Many Glacier (a resupply point with store, motel, etc.).
There is not a lot of car access to the park. It is divided north south by the Going to the Sun Highway, which starts in St. Marys and ends at West Glacier. The CDT crosses this highway at two different points, depending on your route. You can probe the east side of Glacier NP at three points - Many Glacier in the north, at St. Marys in the middle, at Two Medicine in the south. East Glacier is the town at the far south east. The train goes through West Glacier and East Glacier.
The shuttle systems can get confusing. The park service runs three free shuttles across the Going To The Sun Highway. There is a shuttle for the west, middle, and east, so it takes three rides to get all the way across the Going to the Sun Highway. They do run frequently - every hour or so. See the park service schedule. There are also for fee shuttles running up and down the east side, as well as pay shuttles to connect to to the nps shuttle at St. Marys from Many Glacier or Two Medicine. The cost was $10 for each link, i.e. East Glacier to Chief Mountain $40 per person, on to Prince of Wales Hotel, $50. There is one shuttle that goes all the was from East Glacier to Chief Mountain. It leaves East Glacier at 11am and gets to Chief Mountain, I think at 1:45pm. They go more often between St. Marys and Two Medicine and Many Glacier. The East side shuttles are on a first come first served basis, so you can't rely on getting on - may be full already. Don't have any answer to that problem. Don't know if it is a realistic problem, or just hypothetical. The bus exists to shuttle train passengers who have a Prince of Wales Hotel destination to that hotel. Excess space is available to hikers.
If you have to leave a car in Many Glacier, you can leave it in the big lot behind the hotel, also, I think, in the lot in front of the Swiftcurrent Motel. We stayed at Whistling Swan Motel in East Glacier, and they were willing to let us leave our car there, as we had a reservation at the start of our trip, and the day we exited Glacier.
If you have a trail angel who can drop you and pick you up in various places, you could slack pack from Many Glacier to Going to The Sun Highway, also from East Glacier to Ahern Pass. They could also drop you at Chief Mountain earlier than the shuttle would..
Resupply - we were able to drop a package at Swiftcurrent Motel (Many Glacier) with the promise that we would pick it up in three days. Don't know if they would accept mailed packages. The store at Many Glacier had freeze dried dinners plus the normal type of things you might buy for resupply.