I found an inverter circuit for a 4W fluorescent light tube. But, it is from Europe and its output is 230V. Living in the US, I would like a 120V output. I know that the transformer needs to be different. What else do I need to change? The power MOSFET and the 100n,1200V cap? Do all the component values need to be re-calculated? The schematic shows 2 options on the output: with 2 transformers and 1 transformer. I want to build the circuit using 1 transformer which they say should work for a 4W fluorescent tube.

 

I guess the 4w neon lamp is a must for you. I would choose a 10mm LED in series from the 9volt source get rid of all circuits and add a resistor in series benefits the 4w watt lamp can be replaced by a superbright LED using maybe 1/2w total without the extra baggage and problems.

 

You will need a 9V/120V transformer. However, I suggest you to use a 12V/120V transformer as not to saturate the core due to the 12V battery.

 

I have never seen a fluorescent light tube that has a power rating as low as only 4W.
A ballast supplies power to a fluorescent light tube. It starts the light with a high voltage then limits the current. I don't think the tube will start lighting with only 120V.

 

Thank you to all who replied.
Alberto: thank you for the distinction in the 2 circuits; that makes a lot of sense. I need to figure out how many volts are needed to start a non-filament tube.
Mik3: I am thinking of following your suggestion of going from 9V to 12V.
Audioguru: they do make a 4W fluorescent tube, the F4T5CW. But, you might be right that 120V will not be enough. I don't know yet.
On the whole, it is good to hear that the rest of the circuit is sound.