Follow up: I'm happy to report that the procedure I mentioned above – a series of forced complete cycles – has solved the swollen battery problem, at least reducing the swelling (though the battery casings are still enlarged, having been stretched by the swelling) to where the lower case could be reattached, and the trackpad works fine. Procedure (h/t Petr Jerabek at Santa Fe Mac Repair):
1. Fully charge battery 6-8 hours, use it with power adapter plugged in for the whole time.
2. Disconnect power adapter, then fully discharge the battery, by starting into either the Startup Manager with option ⌥ key, or into target disk mode with T key (if MacBook has firewire/thunderbolt), and letting it sit until it shuts down. After a few minutes, press the power button; it it starts, let it discharge again until the battery is completely discharged.
3. Let the computer rest overnight (8 hours) in the fully discharged state.
4. Reconnect the power adapter, then immediately reset the SMC (see "How to reset the SMC of your Mac"). (If you're quick enough, the adapter's indicator light won't turn orange until you release the keys used in the SMC reset.) Let it fully charge for 6-8 hours, keeping the power adapter connected during that time (can be used while charging). After it is fully charged, you might check the battery's status e.g. with coconutBattery.
I repeated this procedure 6 times, looking at the battery a few times to see if the swelling had reduced (the lower case was under the computer but not attached, as if it were the trackpad wouldn't click). After the 6th time around, the battery was pretty well flat again, so I reattached the lower case, and have been happily using it for a week now.
As noted above, the battery in this 7 year old MacBook Pro had all of 7 cycles – which I've now learned was a bad idea (I'm working on changing my battery habits), and was at 98% of design capacity; at the end of the procedure it showed 13 cycles, 96%. Just looked now and it's up to 100%. Don't know what that means. I also learned that the battery will last for 2½ hours when discharging in target disk mode (which I'd guess is power intensive for some reason?). Since I use it 99% on my desk, that's not a problem.
I tried the same procedure with a 2013 15" MacBook Pro which also has a swollen battery (somewhat worse and for longer than the 2014). It had no effect. (One of these days I'll get an iFixIt kit and see if I can replace the battery myself.) So it doesn't work on all cases. But if you have a swollen battery that's a recent occurrence, this procedure could save you $200.