Who should be paying for the lease extension on a Shared Ownership property?

what I think is irrelevant

the only thing that matters is what the seller says and how your respond

The seller says they aint gonna pay for it -

so the only thing now is- what are your options?
1) YOU can pay for it
2) you can tell the seller you aint gonna buy unless THEY pay for it

so make your decision and do it
(but bear in mind - the price you are paying probably takes into account the length of time left on the lease
If THEY pay for it then the lease is worth more and they can charge more for the house)

You talk about a 'Shared Ownership property' however then go onto suggest what you are purchasing is a Leasehold and the other party would be the Freeholder.... a shared ownership means you purchase a percentage of the freehold and the other party owns the other percentage of the freehold and that is not for eg '81yrs'

So working on you say you are purchasing an 81yr Leasehold property from the freehold seller and you are the ones who wish to extend the lease years, then you are the one who pays for that, not the freeholder who is selling it

No, I don't think the seller should pay for it.

You are the buyer. Your job is to deliver money to the closing agent to satisfy the sales contract. If you don't have the money then you have to go get the money from a mortgage company and if that mortgage company has rules & guidelines to abide by before they loan you money then YOU are responsible for fulfilling their requests so that they are willing to deliver the money that YOU are responsible to bring to the table.

The seller is not obligated to pay for expenses which are incurred because of your financing.

Consult with an attorney who knows real estate law. You will find you will be saving money and frustration as you continue with this, if you do it properly and with guidance. If you don't, you can expect LOTS of problems, lots of misunderstandings, and plenty of headaches. The smallest detail can produce the largest issue.

Most attorneys will consult over the phone for free for standard questions. If they feel it is more complex, you can then go to the office for a still small fee ($125+ to $200 depending on the attorney). Believe that this fee, if needed, will avoid issues as you continue, and be well worth it. Ask what the fee will be, if this is needed. An attorney experienced in this can advise you on things you didn't even think to ask, or that might arise in this case.