A battery charger is a d.c. power supply. You should be able to find details to build one of those.
The problem comes with controlling it. What you’ve written does not make complete sense. If it’s constant voltage, the current will be defined by the battery - specifically the difference between the driving voltage and the internal battery voltage, and the internal resistance of the battery plus the source resistance of the charger. If it’s constant (or controlled) current, the voltage will be defined by the battery. What I think you need is a current-controlled charger that has a voltage limiter to prevent overcharging. So the main control loop is on current. The battery voltage will rise as it reaches full charge, the voltage limit will come into play and the current will naturally drop away.
You’re talking about 20 A or so in the d.c. side, so you will need fairly substantial active devices (and maybe fan cooling) to handle that.
But personally, I wouldn’t try to build one. I’d look for a commercial unit - almost certainly switched-mode given the power rating you are calling for - with the appropriate output voltage and current, that can be controlled remotely.
I doubt that will be necessary. If your immersion heater has a thermostat, the Mk2 will be unable to divert any power to the immersion heater anyway.