Should I update the Land Registry and if so, how?

TechDad

Evening all. When my Mum passed away a few years ago, she left her 50% share of the house she owned with my Dad (as tenants in common) in trust with my sister and I as trustees/beneficiaries.

We have not completed Form DJP and as such both of their names are still on the title, along with the standard Form A restriction to reflect the TIC.

The house is about to go on the market, and I'm wondering if we should send Form DJP off with a copy of the Death Certificate ahead of time to remove mum's name to reflect his sole legal ownership, and have him appoint me as a trustee to overreach the Form A restriction. If we did this, can we then just transfer the proceeds of the sale 50/50 to Dad's account and the Trust's account, or does the legal ownership need to mention the trustees somewhere ahead of the sale? If so, I assume Dad would need to complete form TR1 to reflect this?

Also, I'm guessing that if we don't get Mum's name removed ahead of time, then we'll need a Grant of Probate to sell the property with her name still on it, is that correct?