Pulling voltage on a "pull-up" input down to match voltage of voltage output from another device
A pull-up works exactly this way: when there is not another voltage source on the section, the pull-up puts this pin at 5v. If the charger is connected to a "dumb" battery (a BMS or battery unable to "talk" to the charger), the pull up will actuate and will set the charger at its peak voltage. 

But if you connect your "smart" BMS unit to the charger, the BMS signal will override the pull-up (a "weak" pull-up resistor of only 10K is feeding the pull up pin) and so the BMS will change the input of the charger, in order to request from it as much current as the BMS want/need. 

I didn't understand why you can't just connect the BMS directly to the charger (and everyone would be happy). I'm sorry to answer your question with another question.

If you really have a good reason to drive the charger with an independent control, or you can't connect them directly, IMHO the best solution would be  an Op-Amp circuit with zero gain, so, you feed the BMS output pin to the input of the op-amp circuit, and connect the output of the op-amp to the charger input; the op-amp has low impedance output, so, it would be easy to override the pull-up. 

This would be a simpler, analog and more elegant solution; but I'm not sure if it would fit, because I don't know the reason why you don't just connect the BMS output to the charger input.

If for some reason the analog solution doesn't work, other option would be microcontrollers; a uC with a DAC (the DAC output should be feed to, guess who, an Op-amp with zero gain), or a uC with a PWM output, which would be feed to a rectifier and after that, to an Op-amp.