Not all trains are long distance intercity trains driving at those speeds over long distances...Bombardier are currently trialling one or two trains with this setup, but at the moment the sheer weight of batteries required to make 300 tonnes plus of train move is...well, a hell of a lot. I can't imagine it being cheap either, and I can't see batteries powering trains at 125/155/185mph including all the acceleration (regenerative braking already exists on trains) for too long. You'd need a hell of a charger.
The Diesel electric train I used to catch into Glasgow Central peaked out at about 65 mph, (measured with GPS) and had a 50 mile round trip from the main station to its furthest destination and back again where it would sit at the main station for a good 20 minutes before setting out again.
Seems like a candidate for a battery / overhead lines electric only implementation. You say you'd need "a hell of a charger", but that's a "we can only charge stopped at the platform" mentality.- what if each stop, plus a couple of miles of lines on either side of each stop was electrified but the main portion of the line out in the intervening countryside wasn't ?
As long as the ratio of charging rate and charging line length vs discharge rate and total journey length was sufficiently high it would work - no need to electrify the entire line.
Something similar is already being done with buses with overhead inductive charging at each stop - it would be easier to do it in the train scenario to be honest, as most lines are already at least partially electrified.
By the way are you sure all Diesel Electric trains have regenerative braking ? I've had a pretty good look at the generator cars on the one I used to catch and didn't see anything looking like batteries. Another dead giveaway is that the Diesel engine always revs up to high revs before the train starts moving then throttles back once it's up to speed.
If it had regenerative braking (and/or any charging of surplus energy from the generator into a battery) it should have enough energy stored in the battery to accelerate the train from a standstill without revving up the Diesel generator, and just use it like a Rex. But from observing the behaviour of the trains day after day this is not happening. The Diesel engine is always revving up just before some acceleration.
I don't believe any of the Diesel Electric trains I caught from Glasgow central had regenerative braking, or indeed any batteries. Just a directly supply from the generator set to the electric motors and normal friction brakes for stopping.