I'm not sure about the circuitry you're dealing with, but it may be possible to get it to run (not at it's best, but run) with less than 1 amp. If so, you're still looking at making a device that will convert the DC to AC at the same frequency produced by the transformer. If that is the case, you're looking at creating an inverter. It is possible to make a simple one with just two small (.22 uF capacitors), a potentiometer, and a 74AC240 octal buffer chip that will produce a square wave AC output. Unfortunately, such buffer chips at most can only take 7 volts (6 volts is recommended). Thus, if you were to use this method, you'd have to route the output to transistors that will in turn be able to switch on and off the heavier power load.
As a side note, a circuit utilizing the more common 555 timer can be made to produce an AC output, however, careful calculations must be made regarding the size of the resistors for it to produce a near 50% duty cycle (when the alterations between + and - are equal).