Rhumblines aren’t straight lines (with 2 exceptions) so you’d definitely want to travel along a geodesic / great circle. Latitude lines (except the equator) appear to be straight but are curved ) The 2 exceptions traveling along constant headings (Rhumblines) are along the equator and the meridians.
Cesium geodesic methods lack the ability to determine a geodesic if given a point, distance, and initial heading. https://cesium.com/docs/cesiumjs-ref-doc/EllipsoidGeodesic.html
-You can feed it 2 points to get initial and final headings.
-You can interpolate,extrapolate a geodesic to get another point along the geodesic.
On this thread
Ellipsoid Geodesic methods
There’s a link to formulas that’ll give you the formula you need
I transcribed the one for spheres from LatLon site. For most purpose it will be close enough when applied to ellipsoid Earth. The one for ellipsoids is alot more involved (but very accurate), I might transcribe that one at a later date.
(removed code from here as I have it on my next post)
What you could do is get the destination point using the transcribed function I placed here, create a geodesic with the original start point and use this to get the end point. Then compare the initial heading. I bet it will be close enough.