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I'm in a very similar property (Victorian house, I'm in the upstairs flat) and own a share of freehold with the downstairs. We don't do it through a limited company (not really sure what the benefit is for two people and the faff of having to file accounts), we're both named on the freehold title and then each flat has a lease.
I get on well with the guy downstairs so it works fine. The only cost is the building insurance, maintenance costs are dealt with as and when they arise. It's an easy arrangement and you have the security of being able to veto any stuff you don't like.
One big benefit for me was that I could do a loft conversion with a minimum of effort (just had an email from the guy downstairs saying he was OK with it) and extending the lease is just solicitor's fees.
I'd agree that share of freehold would probably add more than £3k to the property value, it's just a bit more security than only having a lease.
Owning the whole freehold gives you more options but a share is definitely beneficial.
I'm not sure it really would as the lease is already 125 years. It probably would anecdotally. I can hear the estate agent now "share of a freehold" wink wink