Its been done before, although I thought we were already getting SM-6 and it was SM-3 we needed for BMD. That said both SM-6 and ESSM (especially the active Block II version) are intended to be part of the USN terminal BMD solution to counter the DF-21 area denial weapon.
BMD covers a broad range from shit that is literally right on top of you, to space intercepts, to getting things in boost phase. Ideally you would have the right weapon for each of those.
SM-6 is pretty capable. But SM-3 is in a completely different league. ESSM is going to be able to mop up a direct attack, but you are probably not going to want to rely on it as your only defence. Realistically SM-3 is the most likely to be the weapon that stops a BM, with the range and altitude.
Given the type of missiles we might want to stop, anything holding SM-3 is going to be a lot more useful and capable at controlling airspace. Anything with Sm-6 and ESSM, unless they are being directly engaged by the BM is just going to watch as it goes overhead.
If we were to build 3 more F-105 based ships, it would seem likely that we might pony up the money to make them full (and latest) Aegis ships. 3 AWD really isn't ideal, it never was. 4 is at least plausible (ala the 4th AWD which was planned but not funded in the beginning), 5 or 6 is reasonable (like Spain or pretty much any Aegis fleet). With 3 you just don't have the numbers to have an independent capability or full use of its advantages as you are basically talking continuous availability of 1 ship, which given the other issues (transit etc) is not going to be that useful for anyone.
Actually I would plan to make 6 more AWD based ships, have them with the latest baseline BMD setup. Two of the AWD are then sold to NZ (or say Singapore) replacing the Anzacs (or the Singapore corvettes). Perhaps NZ would take two, and Singapore would take one. Joint training, logistics and deployment with Australia leading that, but all three having significant roles to play.
That starts to have a larger impact, Japan only has 6 Aegis ships, so being able to bring another 9 (admittedly smaller ships) into the region shifts things much more significantly than Australia operating 3 far away independently.
It also ties Singapore/NZ and Australia much closer together. I know this idea would be highly controversial in Singapore, NZ and Australia, but its the sort of collaboration that should be happening within the region. The US would have to agree obviously, Im sure other players in the region will eithe feel left out or against it (sprinkle the 10 anzac hulls around).
Australia could then continue its rolling build, perhaps additional ships to NZ/Singapore (NZ takes 1 extra to make 3 and Singapore takes one more so 2), with the Anzacs going to various friendlies within the region.