The 3 AAA string probably has some type of voltage reducing component(s) to lower 3 x 1.5 = 4.5V down to the nominal 3V operating voltage of a white LED. So if powered by your 3V adapter, I'd guess it would be dim or perhaps not light at all. See if your 3V adapter says "regulated" on it somewhere. Some 3V adapters are "unregulated" and put out close to 4.5V when lightly loaded and only drops down to 3V at some rated "full" load. So with the combo of items you've mentioned, you might find one or the other works but trying to power both will change the brightness of either individually.
If there is a problem with voltage mismatch I suppose you could mess with components (adding voltage dropping components or removing the voltage dropping components from the 4.5V string) but it's probably easier to buy another matching string.