Ferdinand, this is incredibly helpful advice - thank you for taking the time to lay it out in such detail. I've had a scan through the links and as you say the FAQ on consent is very relevant to our situation.
By chance we spoke to the leasehold advisory service today. You have to book a 15 minute slot online and they book out fast so we had been waiting a couple of weeks for advice. Interestingly, they gave similar advice to you - essentially that consent could be deemed an implied clause, even if not explicitly stated in the lease. The key points seem to be whether the external walls and gardens are included in the lease demise and also the general wording of the lease. Interpreting leases is obviously a very technical area and ours seems ambiguous even after a read through of all the relevant guidance. Luckily the leasehold advisory service will also review written documents and provide free advice in writing so we have duly sent our lease and plans off to them for a second opinion. The only drawback is a further 2 week delay - but obviously worth it if we can avoid or reduce a £5k premium.
The next difficulty seems to be enforcing our rights, even if they do exist. Freeholders are in a strong position here, because building a case that consent is being unreasonably with-held will be almost as expensive as proceeding with the freehold purchase, once surveyor valuation and solicitors fees are taken into account. It's definitely a minefield, and one we could have done without, all else told. Our predicted cost for adding 12 m2 of space, a new kitchen and general refurbishment is now up around £70k... it's absolutely horrendous really. When I have a moment, I'll post in the extensions section as a cautionary tale for others.
Anyway, I'm sure this is fascinating for everyone else on the Buildhub - it certainly has been a unique learning adventure for me! Looking forward to getting into some of the more practical construction challenges.