How to figure out timing for working on and submitting independent research?
I can not give you a fully fledged answer at all, but some things to take into consideration:

Where do you want to place your article? This determines the amount of time you will have to spend on it. This has to be in-line with your idea about being short on papers (because it might not help the issue if noone reads the journal).
I personally would find it a tad rude to publish alone, if the idea has spawned from something connected to your work for your insitute (and might also not be allowed depending on your contract).
If you are not pleased with option 1 with respect to your ideas about the paper's benefit for you I would go with 3. You could also think about finding a PostDoc position which includes the work on this subject. If you do this intelligently, you could also use this as a selling point for yourself. Ideally this also would be the general idea of a perfect PostDoc position, allowing to develop your own ideas and research. You could try doing so with a research stipend while going abroad.