Moving the building would, I'd expect, cost a lot more than the cost of removing the pylon.
It's a very old house so there'd need to be work to check it was sound enough to move, so specialist structural engineer fees.
Probably some kind of permit to move the building (or just plain old planning permission) and any associated legal fees with that (even if it's on your land, it would change the deeds etc).
New foundations dug.
New utility lines put in and re-routed (so plumbers, electricians, gas engineers).
Groundworks to prepare the ground you're planning to move the house too - check it can take foundations, has no surprises underneath, etc.
As it's an SSSI I'd imagine there'd be some requirement to have an impact study or surveyor from that angle, and I believe you'd need consent for any works of this scale.
A path cleared for the hydraulics platforms to be able to move the house - is the site clear and able to take the weight of the house along the entire route? Again, I assume this would require a land surveyor or structural engineer of some kind.
Then excavation work is done on the existing property site to cut around the foundations, if there's a cellar that complicates things, and it's put on the hydraulics and moved to the new foundations, everything is reconnected, etc. This part alone will be expensive, and it's not commonly done in the UK, so I imagine the price will be more as it's a specialist service.
Then there's the potential for damage, the house settling weirdly on the new foundations, etc... you'd have extensive insurance for it I assume (again, probably have to be specialist insurance!) but still, it'd be a hassle.
Having said all that, it's a very cool idea and I love seeing how buildings get moved (it's a bit of a weird special interest of mine as I work in property)! But it's an enormous amount of work and cost for an individual, rather than a business, to bear.
You might have better luck paying a very good lawyer to try and find a reason for the energy company to have to move the pylon!