That's a very interesting question!
We don't have an oblique version of the Lambert conformal conic which would be the best match. My next thought was possibly Hotine oblique Mercator or rectified skew orthomorphic (same but latter has a rectifying angle parameter). My next thought was to try the Krovak projection. It's an oblique conic, conformal that was designed for Czechoslovakia and is still used in the Czech Republic. The parameters are a bit of a mess, but I quickly threw together some that, to my eye, look like it's approaching (very roughly!) the picture in Snyder's Map Projections: A Working Manual on page 121.
bipolar_equivalent
Authority: Custom
Projection: Krovak
False_Easting: 0.0
False_Northing: 0.0
Pseudo_Standard_Parallel_1: 30.0
Scale_Factor: 1.0
Azimuth: 50.0
Longitude_Of_Center: -20.0
Latitude_Of_Center: 45.0
X_Scale: -1.0
Y_Scale: 1.0
XY_Plane_Rotation: 90.0
Linear Unit: Meter (1.0)
Geographic Coordinate System: GCS_Sphere_ARC_INFO
Angular Unit: Degree (0.0174532925199433)
Prime Meridian: Greenwich (0.0)
Datum: D_Sphere_ARC_INFO
Spheroid: Sphere_ARC_INFO
Semimajor Axis: 6370997.0
Semiminor Axis: 6370997.0
Inverse Flattening: 0.0
Don't pay too much attention to the GCS. Snyder says that a sphere was used, but doesn't say which one. If the data/map does, you should use that information. A sphere with a radius of 6370997.0 is approximate the authalic sphere for Clarke 1866 (same surface area).
Melita