Voltage Doubler stops 'Doubling'?
Keeping the left hand cap at 1uF is the problem. Just do the math - it has an impedance of 398 ohms at 400Hz and in your original circuit you had a 100k load. No problem here but, dropping the load to 1kohm means you should increase the left hand capacitance appropriately in order to counter the loading effects. Or increase the operating frequency in order to reduce the 1uF reactance.