HSTEd is there any question where your answer isn't third rail electrification?
Third rail doesn't solve enough problems, and it creates new ones. There is the obvious concern about safety, especially as the bridge is a special structure where normal maintenance access arrangements do not apply. New third rail installations for metro systems come with access routes to the side which mean that maintenance staff and evacuated passengers have little reason to come into contact with the third rail regardless of the state of the system. I can't see it being possible to add a compliant access route on the bridge.
Also, third rail DC just means you swap one set of electrical compatibility issues for another. Even if you have the physical clearance for the third rail and the train collector shoes, you have to worry about how you prevent stray currents ending up in places they're not meant to be. London Underground had to work out third rail DC in a metallic environment and they came up with a four rail solution. If you would need to do something like this, you'd be going for a unique system again.
Batteries are the best solution. In the worst case scenario where there is precisely zero clearance, an EMU can drop its pantograph and have the same loading gauge as a DMU. The battery means you can have inclines, stations and signals in the dead section which would make coasting impossible. Where there's physical but not electrical clearance, you can have neutral wires or conductor bars to avoid the need to raise and lower the pantograph. There's no need to do any more infrastructure changes than is absolutely required. The work is all on the rolling stock side. All of these options, including third-rail, require some special rolling stock treatment. Batteries would be one thing that would be universally useful across the country's EMU fleet, so there's no need for ROSCOs to worry about assets which are only useful in one setting. The battery-equipped trains running from Edinburgh to Dundee could later end up used to extend EMU services just beyond the end of the wires elsewhere.