I have had a quick read over your thread. Bear in mind that we only get a fraction of the story here, so it's worth at least having a consultation with your lawyer about this since they know the whole story.
The majority of costs orders made in relation to parenting proceedings are the result of some additional proceeding, rather than the initial proceeding, for example an application in a case, an application to vary orders under Rice & Asplund, an application for contravention orders, or an appeal. It's extremely uncommon for costs orders to be made for the initial matter,
Based on your first post about costs, I'm inclined to assume your main concern is about the 'offers to settle' consideration under s 117. The ex might have made dozens of offers to settle, but unless you had an ICL and a family report writer making the exact same recommendations as what was repeatedly proposed and then made in final orders, this maybe won't be much of a determining factor (less so if, as you suggest, the father was clearly, not just speculatively, linking property to parenting). Remember, as well, that you were both privately represented, as well, so unless your conduct caused unnecessary additional legal fees for him, you're both in the same boat.
So, I still don't think you should agree to pay costs without a Court order to do so, and rather than spending more money on a lawyer to debate costs, I would probably get one last consultation for legal advice, then consider dismissing your lawyer's services and corresponding with your ex's lawyer directly as an SRL, at least until you've got an actual application for costs sitting on your desk.
And remember, your ex's lawyer knows that just as they can apply for costs, so can you...