You can treat your cold frame like a raised bed, and fill it with a bagged "raised bed mix." I don't know what's available in your area. We have plenty of suppliers to choose from here in the desert, not sure about other parts of the country.

Yes, pure compost is rarely recommended for planting seeds or even starter plants and for more reasons than just the weed potential. You really want a mineral based soil....like garden soil (with some organic amendments) or an appropriate bagged raised bed soil.