Isn't the sea-wall and cliffs at Dawlish, and the very railway itself, of special landscape value? Masts and portal structures (lattice ones, I read, with special debris screens), at closer spacings too, and contact wires and catenary (2 of each, of course) would mess up the landscape somewhat, don't you think?
No different to Sydney Gardens in Bath, the Royal Border bridge at Berwick upon Tweed, Edinburgh Waverley station or countless other projects. Edinburgh Waverley and Sydney Gardens are in UNESCO World Heritage sites, and the Forth Bridge, which is in line for electrification, is similarly a UNESCO World Heritage site.
The absolute most you can hope for is some architecturally sympathetic masts onto which the prevalent OLE system of the day will be mounted upon, though for most of the route, I would probably suggest twin track cantilevers mounted on the cliff side of the route would be both sensible engineering solutions and will have the least impact on the landscape.
I do seem to find myself repeatedly reminding people that a far more historically important and significant railway, the Liverpool and Manchester Railway, has successfully been electrified, with stations including Rainhill being sympathetically treated.
Those hoping to play the landscape or architecture card to stop electrification really need not bother - focus your energies on ensuring the work is done as sympathetically as possible, suggest practical and affordable options when the Network Rail and contractor teams conduct public meetings, explain to them patiently where you wouldn't want a mast or portal and suggest alternatives.
It's worth bearing in mind that the designers, consultants and engineers who draw up the plans won't know your town or village intimately so if they're missing something, kindly point it out to them instead of assuming they already know and don't care.