OK so generally, the non-primary carer finds themselves with alternate weekends... Not you.
My thinking - forget solicitors. Try and get agreement, be patient, ask for one weekend night a month, even if it is a trade off for a different night some other time.
So obviously you've done mediation? I'd be patient. But before the 12-month expiry date of the mediation certificate, apply to court. Look, eventually you're gonna need court orders / consent orders - otherwise you're at her beck and call, and you'll have no certainty about when she might change her mind again...