I'm pleased to hear that.
That means that the court has the same powers to make orders about property and finance, and that you have the same legal rights, as a straight married couple who divorce - but those rights and the court's powers only come into play once you have filed a petition at court to dissolve your civil partnership.
I can reassure you that for the property to be sold, your ex either needs your consent or she needs a court order - but the court would consider all the financial and property aspects of your partnership, not who shouts the loudest. The housing needs of the children will be a priority for the court.
Once you have filed your petition, either one of you can then make a application to court to ask the court to decide how the assets of your partnership should be divided between you, if you and your ex cannot agree between you. You will have to pay your ex SOMETHING to buy out her share of the property - but not necessarily 50% of the equity.
The court starts from the position that all the assets should be divided 50:50 - but then looks at reasons why that should not be so eg if one party has a significantly lower income than the other and/or is providing a home for children of the family - then the percentage shares could be 60:40, or 70:30, etc.
If you can negotiate an agreement, whether by discussion between the two of you, or via solicitors' correspondence, or via mediation that would be preferable, as going to court is stressful, time-consuming and expensive. The family court anyway now requires the parties to have attempted mediation before it will consider an application to court.
Here's where to find a local family mediator:-
I think you would benefit from some face-to-face legal advice.
Here's where to find a specialist family law solicitor:
I hope this helps and I wish you the best of luck.
Thanks and best wishes...