Transistor getting hot in rail splitter circuit
This is something close to what I'm thinking about. It's really not all that different from the usual class-AB power amplifier stage of an audio amplifier.







simulate this circuit &ndash; Schematic created using CircuitLab

This circuit includes some details to handle BJT variations and using the Sziklai arrangement helps circumvent issues related to heating in the power BJTs, \$Q_1\$ and \$Q_2\$. You will need to adjust \$R_{14}\$ to set the quiescent current (just measure the voltage across \$R_1\$ or \$R_2\$ while setting this.) I've also included \$R_{13}\$ to handle the Early Effect of \$Q_5\$ and \$R_{16}\$ to deal with accidental shorts (you can remove it, if you want, by simply bypassing it with a wire.) \$R_9\$ and \$R_{10}\$ are there to avoid oscillation and to help match BJT vagaries. But they also can be removed (bypass them) if you want.