you don't want to remove 100% since you want to charge the batteries somewhere. Also it maybe more economical to keep 1km of power lines after the station when the train is accelerating - main power consumption.
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That's a great idea. You could just put the overhead charging infrastructure at a few stations on the line. Just enough to top the batteries up. Most of the energy could probably be recovered through regenerative braking, but of course some will be lost through mechanical and air resistance.
This would be so much simpler than electrifying the whole line and much easier to maintain.
This infrastructure is often already at key stations, and so it might require no extra work to implement them. Just more money for the battery powered trains.
Gotta put little spinny generator fans and solar panels on the train too. Perpetual motion!
edit: also I don't think your idea will 100% work because it would take some time to charge the trains at the stations.