MHG

There´s one connection for 70,-€ ("69,90") availlable which gets you to Heinebach 3 minutes later than the more expensive one.
The "trick" is to untick the field "prefer fast connections". The result is that you´ll change trains already at Fulda rather than Kassel ...

Both connections mentioned so far leave Stuttgart Airport railway station at 19.08
Which leaves only 33 mins between scheduled flight arrival and train departure - that´s pretty tight.
If you have only hand luggage you´ll make it. With checked luggage it´s not possible even if your flight is on-time.

... and this 19.08 departure is the last one that let´s you arrive the same evening at Heinebach. If you miss it you´re stuck.

The only alternatives:
I checked with Flixbus and there is one bus which suits your plan/requirements only in part and works only if someone can pick you up at the bus stop in Fulda as there´s no train to Heinebach before 06.16 going from Fulda to Heinebach.
This Flixbus does not leave from the airport but Kornwestheim which is reached by the same local train (S-Bahn) that would be part of the expensive train ticket from the airport.
Going to Kornwestheim requires a transfer at Hauptbahnhof (Stuttgart central station).
Total travel time airport to Kornwestheim will be 42-52 mins and tickets cost 5,30€
The Flixbus leaves Kornwestheim at 21.05 and arrives Fulda 0.25 (Busticket is only 9,90€ at the moment)
At least this option would give you a bit more time at the airport since the S-Bahn runs in 10-20 mins intervals.
S2 + S3 go from the airport to Hauptbahnhof (11 stops) and S4 + S5 go from Hauptbahnhof to Kornwestheim (4 stops)


... and finally if you give up on arriving on saturday night or stay over night in a hotel Flixbus has one bus straight from Stuttgart airport to Kassel (29,99€) leaving the airport at 23.05 arriving at Kassel Wilhelmshöhe 05.40. That would be perfectly timed for the first train leaving Kassel Wilhelmshöhe at 06.14 arriving Heinebach 06.46
In the end you´ll pay for that trip in total 39,60€ (29,90€ Flixbus Stuttgart airport to Kassel and 9,70 for the train Kassel to Heinebach)
That´s considerably less than for the train ticket for 69,90€ but requires you to travel overnight on the bus and you arrive at Heinebach sunday morning 6.46 (pretty "ungodly" time for a sunday morning to arrive - if you ask me - when everybody else is asleep ...)