As i say if the operating mode is going to be to always use the batteries for acceleration, even when on 3rd rail, then overall energy efficiency will be improved as your guaranteed to have somewhere to recover the energy lost in braking especially as regen is really only effective where traffic is high density. Also the problem with DC systems is when they are beefed up they come become less receptive as average voltage at the shoegear will be higher. Im not aware of technicalities of the PS upgrade but i'd be surprised if its higher than the "high current" standard on SR which was 6.8kA as that was designed to support a class 373 Eurostar (it was gutless mind you [on 750V DC of course]).
If only the battery was used for acceleration and only charged from the third rail, not braking and if no power was drawn directly from the third rail for acceleration, the system would be hugely inefficient. Only if the battery is predominantly charged by regenerative braking, with charging from the third rail only used to improve acceleration while reducing the load on the third rail, or when the unit needs to run off the third rail, but not under the wires, except to Preston where this requires running over unelectrified infrastructure to reach the overhead wires, is it possible for overall efficiency to be improved.
The full technical standard for the third rail on the SR allows for a certain amount of current at 750V, but nowhere on the SR is it actually implemented. The Merseyrail upgrade still does not exceed this SR standard but the installed current capability is now higher than anywhere on the SR. I forget the exact values but they are in a nearly 2 hour long video with David Powell.
Conceptually it doesn't make sense to run under the wires but then lots of things don't on todays railway. The additional load on the traction system depends on the state of charge of the battery and how quickly you want to recharge it. The trains are nominally rated at 1.5MW with 2.1MW quoted as peak power so whenever the train isn't motoring that capacity can be diverted to charging along with whatever you get back when regening. Well designed software can easily managed the situation.
The hardware may well be able to deal with the additional load, but if the purpose of the batteries is to reduce the load on the traction system, running on battery under the wires doesn’t make sense unless you have to run over unelectrified infrastructure to reach them, in which case the battery is more efficient than Diesel. I’m sure the battery charge rate can be controlled in software to ensure that the battery isn’t too full to accept regenerative braking energy but is always full enough not to strand the train when running off the third rail. 1.5kW and 2.1kW should be the input power to the motor. A single train should be able to draw more power than that to power auxiliaries including battery charging but you wouldn’t be able to sustain many doing that at once.
I actually believe Merseryrail have some vision here about what could be done with these units but with the way things are currently not sure we will see it realised anytime soon unfortunately.
The main priority does remain being able to replace the virtually 45 year old 507/508 fleet with what should be much more capable modern units that have already been built, but this has so far proved impossible.
I don't think you need much vision to work out what could be done with them, as it's fairly obvious - Preston, Wigan, Helsby, Wrexham and Warrington are basically the options, plus the less likely one of Southport via Burscough South Curve. There aren't enough units to do them all, though, and Warrington (unless it was segregated and the other side Metrolink) would have the serious disadvantage of importing Castlefield unpunctuality, so politically we will have to see.
There are questions of going beyond Preston, Helsby and Warrington, but the problem with the CLC is the number of additional non-stations other than Irlam east of Birchwood. Liverpool - Ormskirk - Southport is currently looking one of the most likely, following its inclusion in Steve Rotheram’s map. The reality is that there aren’t enough units ordered to run any extensions at all and that 52 wasn’t considered enough even to run Headbolt Lane, so any potential extensions will warrant the procurement of further units.