Hi Zach,
If your original transmitter was properly designed, you would be able to simply divide some resistor values by 3.
For example, the base voltage of Q1 must be at least 0.7V. R2 and R3 form an 11:1 voltage divider that provides Q1's base with only 0.82V with a brand new 9V battery and actually less, since Q1's base current will load it down and reduce it.
When the battery's voltage drops to 6V, the base voltage will be only 0.55V or less and Q1 won't work! Will it work with a 9V Ni-Cad rechargeable battery that actually produces only 7.2V? Probably not. Poor design.
If you use resistors with 1/3 the value for R2 and R3, the voltage divider will be the same. It won't work.
In your new transmitter circuit, the voltage for Q1's base is provided by feedack by R2, so it remains fairly stable with supply voltage changes.
Its emitter resistor for the oscillator is 1/3 the value of the 9V transmitter so its Q2 current will be the same at 3V and it will have the same range.