First a general comment - everything in the world is described by either classical or quantum fields. Point particles are a fiction, sometimes useful, sometimes not. Starting with classical field theories like Maxwell equations or general relativity, you find that you are forced to forget about point sources and repalce them by continuous charge or mass distribution, otherwise you get all kinds of nonsense (non-locality, acausality, etc. etc.). One of the reasons for that is the infinite self-force or self-energy problem that crops up already at the classical level.
We can approximate a continuous distribution of matter by a "particle" if it obeys certain conditions, roughly speaking it has to be localized and weakly interacting. By "localized" I mean that all relevant observable quantities (expectation values of operators) are localized. This is not the situation you describe - the wavefunction is approximately localized (with two centres) but it is not observable. Relevant observable quantities like the expectation value of currents will not necessarily be localized.
So, what you are asking in effect is the self-force for a particular distribution of mass (or charge). There is an answer for that, but since you are asking a question that had to do with short distance physics, the quantum mechanics of the gravitational (or electromagnetic) field comes into it. Probably not enough space-time to elaborate on this here.