I think you need to reconsider your request.
First, what is the output current range you need and what regulation level (how much can the output vary with a change on the input voltage or output current) do you need to maintain?
Second, do you need more than 9 volts output when on battery? If so, then a regulator won’t do; you need a full DC to DC converter and step up transformer to get the higher voltage. The alternative would be to go to a 36-48 volt battery.
Third, if you intend to connect to the mains, a transformer is desired just from a safety standpoint. Given item two, it could be combined as a three winding unit.
Fourth, does this thing have to charge the battery? If so what chemistry and capacity is the battery? In this case, it makes sense to add another winding just for the battery charging circuit.
Last, with a need to drop that much voltage when on AC, it probably makes sense to go two-stage. An SCR phase control preregulator followed by your switch-mode post regulator. While this can in theory be done with a direct connection, I have never seen it done that way. That fact alone should tell you that attempting this without a transformer is a bad idea.
All in all, you would probably be better off just buying a UPS.