Debbie from North Bay, Ontario has electric heaters sunk right into her floor. Very few people even know that these things exist, but if you have a baseboard heated house you know how hard it is to get heat in front of a patio door or some other place that needs heat but there seems no way to get it there. Follow this link for more details on the heaters themselves and where to get them.
In the first graphic you see the heater itself pulled out of its box, the box installed right in the floor. The air doesn't come from below but falls into the box on one side and blows back out hot on the other side.
The second photo shows you the metal box itself. The whole heating unit comes out easily to allow for cleaning the box of stuff that falls into the grill.
For Debbie this box is a source of cold air when the heater is not operating. I would suggest doing some air sealing on the wall next to the box in the basement. That should stop part of the problem. Then use high temperature stove or muffler cement to seal up any holes in the heater box itself, to stop all possibility of cold air from below flowing through. Then you can simply stuff fibreglass around and under the metal box, maybe even use a wire mesh to hold it into place. Don't use a vapour barrier at all as the sealed metal box is in fact the vapour barrier, and it is on the warm-in-winter side.