I don't now if your simulation is correct, but you could make it less beta-dependant by increasing negative local feedback, by for example emitter degradation, such as increasing R1 and R2.

By using a Darlington or Sziklai pair, R4 and R5 can be increased while still maintaining output current. Discussion of thermal properties: http://sound.westhost.com/articles/cmpd-vs-darl.htm#s4

If some crossover distortion is acceptable, the biasing voltage could be lowered so that the quiescent current through Q1, Q2 is 0.

I don't now if your simulation is correct, but you could make it less beta-dependant by increasing negative local feedback, by for example emitter degradation, such as increasing R1 and R2.

By using a Darlington or Sziklai pair, R4 and R5 can be increased while still maintaining output current. Discussion of thermal properties: https://sound-au.com/articles/cmpd-vs-darl.htm#s4

If some crossover distortion is acceptable, the biasing voltage could be lowered so that the quiescent current through Q1, Q2 is 0.