Here's how several section hikers I know have done it.
1) Park at Jo-Mary Campground, accessible part way thru the 100 Mile. Have your resupply for north of there IN that vehicle.
2) Get a shuttle from Jo-Mary to Monson, packing what you need to get from Monson to Jo Mary.
3) When you get to Jo-Mary Road, hitch to the campground, spend a night (or more), shower, eat well, resupply from your vehicle.
4) Drive to Abol Bridge Campground, park nearby (free parking right down the road). Have your resupply for north of there in your vehicle. While there, reserve a campsite for your anticipated arrival date back at Abol.
5) Shuttle back from Abol to Jo-Mary Road.
6) Hike to Abol Bridge from Jo-Mary Road, spend a night, clean up, resupply from your vehicle. If you are sure you're entering Baxter the next day, hike a half-mile or so north before going to sleep and put your name on the list for The Birches next day--sign-up sheet located at the info kiosk right on the AT near the Baxter SP boundary. (Max 12 allowed per night at The Birches, so signing up early is advised.)
7) Next day, hike Abol to The Birches via the AT, signing in at Katahdin Stream CG office enroute. Spend the night there.
8) Get up EARLY, hike to Baxter Peak and back, then take the Blueberry Ledges Trail back to Abol (half the mileage as the AT).
9) Upon arrival back at Abol, you have options. If there is space available you could stay at Abol Bridge Campground again. If not, there is another more primitive place to camp down the road. OR, you'll have your vehicle parked nearby and you can drive anywhere you want.
Only foreseeable snafu with this plan is, of course, weather. You may need to stay more than one night at Abol the first time. Baxter SP only lets you stay one night at The Birches, weather issues be damned. It is tough to get "regular" accommodations at Katahdin Stream CG without reservations FAR in advance. So you want to make sure (well, as sure as weather forecasts can be) that one night at The Birches precedes good weather the next day for your summit. If Abol is booked up on a second or third night, you could try other camping nearby or since you'll have a vehicle there you can drive anywhere within reasonable distance to get you back early enough to Abol whichever day you want to hike into Baxter. NOTE: The hike between Abol and Katahdin Stream CG is mostly very easy.
This involves a lot of driving and potentially expensive shuttles. It is anything but a straight shot driving between your three points (Monson>Jo-Mary>Abol). For your shuttle between Abol back to Jo-Mary, definitely compare on-ground shuttles to the air taxi. I've heard it is less expensive.
If you are fortunate enough to have two vehicles, you can shuttle yourselves, tho be prepared for hours of rural Maine driving x 2.