It looks like you didn't yet start to analyze the behavior of your VCO design, otherwise you should be able to answer the question yourself.
No matter if you use a charge pump, or other circuit elements, you set up a control loop. The circuit between the phase detector output and VCO input defines a transfer function. A "wire" would be also a kind of transfer function, if it's suitable depends.
As already mentioned, you need a non-zero transfer function gain at DC to achieve PLL lock. The said "DC-bias" problem might require a level shifter. You also need a low-pass filter to convert the pulsating mixer output into a steady VCO tuning voltage. The filter characteristic should be selected so that it achieves sufficient loop phase margin in combination with the integrating PLL characteristic.