A question regarding airfoil parameters
For a wing to be stable in its own right, its aerodynamic centre must remain static with changes in angle of attack. The usual approach is to sweep the wing and wash-out the outer section. Because your wing is unswept, you have only two choices in creating a stable aerofoil:

A symmetrical aerofoil, i.e. with zero camber.
A cambered aerofoil with reflex camber towards the trailing edge.

The first of these has poor low-speed characteristics and has really only seen use on aerobatic and early supersonic types.
The reflex type offers better low-speed characteristics, thanks to its positively cambered leading edge. RAF 34 provides a good example; it was developed in the 1920s when RAF 33, with even greater reflex, was found to have high drag. RAF 34 had much better performance above 200 kt and was used on several tailless types (usually swept and washed-out as well) such as the Westland Pterodactyls. It would be a good starting-point for your search.
But be aware that your tapered &quot;plank&quot; wing will have abysmal handling characteristics and a vicious, unrecoverable stall. Washout can help a little here, and the CG position will depend on this, as well of course as on your chosen aerofoil. Maybe using it as a hang-glider might be tenable, as the pendulum stability will mask the worst handling issues, but I would not want to take it up in case it stalled!
Addendum
Some remarks have touched on the NASA Prandtl-D experiments led by Albion Bowers, which owe much to Prandtl's &quot;bell-shaped&quot; lift distribution and Reimar Horten's exploitation of it in his flying wings. Prandtl's analysis applied principally to the straight wing and was not concerned with stability or yaw. By contrast the Horten/NASA implementation is applied exclusively to tailless swept flying wings, where stability and proverse yaw in the turn are essential to docile handling. In this latter context, many subtle variations of aerofoil across the span have been tried over the years. JW Dunne anticipated it all, and more, intuitively back in the pioneer era - the Science Museum archive contains boxfuls of his research models - but he was no mathematician. The precise relationships between the various design features and resultant flying characteristics still remain somewhat clouded and contentious, and Dunne's work studiously ignored, so some aspects of the original question here remain unanswerable. But one thing everybody I have mentioned has agreed on; if you do not sweep your wing then you are making life much harder for yourself.