Amplifier power problems, power supply limitations, or end stage base current draw
I get a nice sine wave with about 6.5V peak to peak... power output
stage constructed with a pair of BC547 npn and BC557 pnp transistors.
It gives me a beautiful clean undistorted wave on the output. However,
when I actually plug the little speaker in, that's rated at 8Ω 5W, the
output waves are suddenly clipped hard at about 2V peak to peak

Consider that the current through 8 Ω at 3.25 V is ~400 mA. Your transistors are only rated for 100 mA absolute maximum, so this is well over their rating. To keep peak output current within the ratings of the transistors you must reduce the peak voltage to less than 8 Ω * 0.1 A = 0.8 V.
At 2V the output current is 2 V / 8 Ω = 0.25 A, well below the capacity of your power supply. The real cause of the clipping is that the op amp cannot supply the required drive voltage to get the output up to 3.25 V, due to internal voltage drop in the bipolar output stage of the op amp (~2 V) and the Base-Emitter junction of the external transistor (~1 V).
This is a common problem faced by low voltage audio power amplifiers. Voltage drop can be minimized by using a quasi-complimentary output stage, and drive voltage can be raised by 'bootstrapping' it from the output. However to do this you would need a fully discrete power amplifier stage.
The simplest way to solve your distortion problem is to just reduce the drive amplitude, which will also get the peak current below the transistor ratings. If you don't care about overloading your transistors then you could use a 12 V supply to get an extra 2 V of peak 'headroom', but you will need to create a 'virtual earth' at half the supply voltage and AC couple the input and output. The circuit would look something like this:-



simulate this circuit &ndash; Schematic created using CircuitLab