You are right it's a HUGE job. But I'm sure it's the heater core giving you the trouble.
Several years and miles ago, this happened to my 240D. It was a slight loss and I kept topping it off for a while. I came across an old radiator and put it in the car temporarily. I then filled the system with BarsLeaks and ran it with BarsLeaks and water for a few days letting it go through several heat/cool cycles. I then drained the radiator and flushed the system well with water and put my good radiator back in. I filled it with Antifreeze and Nalcool from the diesel truck shop, and it's been fine ever since.
I don't know if this would work for you, but I'm not sure I would use the BarsLeaks in a really good radiator, it has pellets that I believe can plug up some of the tubes. It probably don't plug a tube or two in my heater core, but I'm in Texas and have PLENTY of heating capacity with what's left.
I'm not reccomending using BarsLeaks, it's a choice you have to make for yourself. Talk to some others and get some other opinions.
Also your 300 has an auxiliary water pump that it probably would not be good for, if you decided to try my solution, you should probably bypass the aux. pump temporarily.
This is just a thought for you, replacing the heater core in your car is going to be a very big challenge.
'84 Euro 240D, 516K miles
'88 300E 5 Speed
Over 800,000 miles in