The TE70013 has a larger footprint, that might fit, but the pin layout would not match
You can always connect the leads to the PCB by wires.
My AC here in NA is 120 volts. Does that provide more power capacity on the DC side of the transformer?
No, it makes the situation worse.
If I were to have to buy a new transformer at some $20 + to make this work, then I am inclined to just go ahead and get the whole Bugle PS kit. Or just build the battery version of Piccolo, for now.
You can try the cheapest option first, then think about improvements.
Powering both devices from one PS is an elegant solution, but it may pose other problems (interference between the devices for instance) and there is need for sufficient power supply. But you can try - it doesn't require any money.
Bugle PS (or other PS you like) - is an uncompromising variant, but you add another box in the system. Bugle PS is (just my opinion) too heavy to load such small watts device as Piccolo.
Battery - the cheapes variant to try and good for powering very sensitive devices like Piccolo.