6S LiPo/BMS combo charging very slowly
The power supply's output voltage sags down to the voltage of the pack
  while charging it.


If the power supply and battery are directly connected then their voltages must be the same. 

Assuming you had the power supply set to 25.2V (6x4.2V), imagine what would happen if it forced that voltage into the battery. Battery voltage 22.2V, battery internal resistance 7.8m&ohm;, current = (25.2V-22.2V)/0.0078&ohm; = 384 Amps! The power supply obviously can't put out that much current, so it 'fold-back' limits to a very low current until the 'short' is removed. 

Charging through a resistor reduces the current to an amount the supply can put out, so it doesn't go into fold-back limiting. The voltage difference between the power supply and battery is dropped across the resistor. As the battery charges the voltage difference gets less so the charging current reduces. It will work, but the tapering charge current will cause it to take a long time to fully charge. 

Be aware that a battery this size is potentially extremely dangerous. Your BMS should prevent it being overcharged and limit current in a short circuit, but doesn't balance the cells. If one or more cells have anomalous voltages then they or other cells could get overcharged while the total voltage is still within spec. If you don't monitor the cell voltages the first you will know about it is when the pack explodes.

The proper way to charge this battery is with a suitably rated charger that has balancing and cell monitoring. They are expensive, but worth the money for battery health, safety, and convenience.

You can continue charging your battery with your power supply and resistor, but you should regularly meaure the individual cell voltages with a meter or cell monitor/balancer to ensure that they are equal (within 0.03V). Don't charge the pack unattended, keep it away from anything flammable, and have a fire extinguisher or bucket of water (to put the battery in) handy in case the unthinkable happens. A fire retardant Lipo bag is also recommended.