If Oz and NZ want all of that and the tonnage creeps north towards 7000T, then why not simply buy the Burkes and get it over with? Off topic, sorry but the question begged to be asked.
* Burkes are generally more man power intensive. You can design this out, but then you are basically redesigning the ship, hence a new class, hence increased cost and risk. Problems and development costs have to be carried by someone. The RAN is not the USN, and the burke is a big navy ship.
* The burkes I believe are too big to build in Australian shipyards (just). 8,000t I think is our limit full load..
* the F-100 is an already proven design in the water, which aligns fairly closely to what we want. Cost and risk are known. It is already a mini burke (or more appriproiately a burke for a small navy). Some modern designs allow enhancement, hull, radar sits slightly higher than a burke.
Besides this is our frigate. The frigate is most likely going to sit on the same 7,000t hull as the AWD, with very simular weapon systems, but is most likely going to feature a non aegis, but aegis compatable radar and combat system. You get a hell of a punch, for less outlay than a burke (with a lot more local content and build).
So far NZ has not express any public wish to join the Australian superpowerful frigate program.
I think the Australian off shore combat vessel (OCV) is something the UK, US and Au could work together. Most likely 2,000t, with basically a 25mm gun up front (and thats it). But large enough to handle small numbers of troops, rescue a sinking boat of refugees, anti piracy, helos, UUV's, rotary UAV's, ASW, survey etc. Space allocated for Phalanx or something but rarely fitted with them. Something big enough to travel across blue water, but focused on doing its job (which is not air defence or fleet defence).
Essentially a LCS without the high speed requirement made using real today LC OTS tech. This would be a ship I think the US, AU and UK could agree on most of the systems (ie non combat) and design.