An LED will not have the same V-I characteristics as the light-bulb. The purpose of the light bulb is for amplitude-stabilization. You can try doing without it at first to see how the circuit performs in its absense.

You may be able to replace it with another LED/photocell assembly. In this case I think the LED/photocell would need to be used in place of the resistance to ground.

hgmjr

 

Amplitude stabilisation works as follows.
The output is sampled and portion fed back from a potential divider to ground.
One of the resistors of the potential divider is made sensitive to changes in the output voltage. The direction of change is chosen to oppose the output change.
Tungsten Lamp filaments have a positive temp coefficient, Thermistors have a negative one. Either can be employed. An FET can be used as a VDR here.
Alternatively a (zener) diode can be used to limit the output above a certain preset level.
This last and simplest method would be adequate for your purposes.

In the 1970s reasonable quality audio signal generators comprised a Wein oscillator followed by a buffer amplifier. They reached the 1 - 2 Mhz range. The attachment shows a Ferranti design of that era. The top range was 150Khz to 2Mhz.

 

Here is an example of the LED/photocell variable-resistance assembly.

hgmjr

 

I never thought I would so many ideas in such a short period of time ) This forum is very useful