There is no doubt that the 3rd space option will be expensive
There is a precedent, that's not insurmountable. When the disability act came in and the conversions had to be made in schools and public buildings for accessible toilets, changing beds, staff trained up to do personal care, ramps, there were grants made available to LAs from central government, the expectations of amount of provision and speed of it being in place were in line with the size of the setting, the number of people being served and their budget, all new build plans had to contain the specifics from the start. In about five years the provision was pretty standard.
Its very reasonable to legislate that all new build toilets, changing rooms etc, must include gender neutral provision as well as designated sex segregated provision (and from conversation here, the general consensus seems to be towards that gn provision being single contained floor to ceiling cubicles). There should be several designated trans prison units. Maybe a percentage of UK refuges could either be set up or maybe even existing ones asked (with financial incentive) to be the designated placements for gn/trans people/male dv victims alongside provision for women, with male/trans staff employed to support them, so women could choose to accept a vacant bed there if they were happy to but the massive majority of refuges would continue to focus on their core brief of women/children only. Everyone's safety and dignity matters.
The sticking point is that this would mean the government standing up for biological sex being a fact, and while trans people are welcomed into all spaces and equally provided for, no one is viewed as being able to change sex.