Old PI's coauthorship on a paper after finishing the postdoc, a preliminary idea developed then but rejected by him
I cannot see why the former PI should have co-authorship on a paper they did not support or had any interest in. You are not a slave (or at least should not be) when on a post-doc (or any other position) and should retain the freedom to take own initiatives. As long as you fulfil any obligations within the position you are holding, no-one can prevent you from developing your own ideas. I can see an issue if you use materials that involve costs that you are not covering, for example, lab equipment or chemicals, electronic resources that removes capacity or resources otherwise used by the project. I do not count, for example, using a computer and printouts as such resources.

So, for me there is no question you can use the research as your own and you should add only authors that fulfil reasonable contributorship criteria, that is have contributed to the science of your work (see posts under the authorship tag for such criteria).