Your needs don't sound too hard to get, but will take some effort to source parts first and in the hands of the builder. I agree about the twist shifting. I much prefer trigger. One of the parts challenges I see is the want of drum brakes (that are effective). The first thought I had was to search for old SA hubs with drums. But some of them don't have much power and you're towing kids. Not knowing if you have hills where you live I don't know how vital stopping power is.

The rest just takes time and effort from the builder. Making the lugs will likely be needed as the small size and the step through/mixte compound the joints. Filleted joints greatly reduce the trouble. Trailer connection should be straight forward if Burley still offers a axle mounted hitch, assuming the rear axle has enough length to accommodate the hitch. If not then some creative shaping of the stays/drop out design might be needed for their standard connector to work.

There won't be much difference in cost between cro moly and hi tension steel by the time the labor and paint are figured in. However there will likely be some of both used any way. I assume weight isn't high on your list.

Yes, what you're asking for will be a very pricy frame. All the effort and skill that a show bike has but in a configuration that's far less common. You need to find a builder who is excited to stretch their range and has done all the needs before (but maybe not on the same frame). There will be a lot of communication and all the parts should be in the builder's hands before any tube is cut. Expect this to take a long time to complete.

Or just compromise and "suffer" with an off the shelf bike that lacks some aspect. Andy.