OK, so your Li-ion batteries are not "unable to supply" the required voltage or current. It seems, if anything, that they may supply too much rather than too little.
Those speakers say "DC 6 V" on the input jack. Also, 4 D size batteries will supply from 4 - 6 V. One of your Li-ion batteries will supply as much as 8.4 V when fully charged. This is 2.4 V more than the amplifier is asking for.
As I said above, it is possible the amplifier won't mind, and will be quite happy when fed with 8 V or so. On the other hand, it might not. To find out, you will have to examine the circuit. Look at the circuit board, find out what amplifier chip it uses (or does it just use transistors?), look at the data sheets, reverse engineer the schematic. If you don't know how to do this, you could post pictures of both sides of the board. Maybe someone here could help with that.
But unfortunately, unless someone else here also has a Labtec CS-150, it is going to be very hard to say.