Except for the capacitor, your diagram is fine. With the battery, the capacitor wouldn’t add any function.
The core would be irrelevent.
The big issue is going to be balancing the load with your coil. We used to have lots of low wattage incandesant available, but they seem to be disappearing. An LED would probably make a good replacement, in which case a smaller coil may also work, but I’ve never tried that. Its probably been 50 years since I had one of these going, but I’ve seen them in people’s offices from time to time. Everyone seems to have a slightly different approach, but they’re all takeoffs on Tesla’s “free power” concept.
A LED should work even better requiring lower power. Unlike much of the above, this is a device I would expect to work. Have fun building it.