First let's take a look at a commercial kit. This is a converter to use with a car radio.
The signal goes in pin 1 and out pin 4. Where is the diode ring and transformers? They are using a Gilbert Cell.Interesting side story that I will not go into at this time but Gilbert did not design the cell? Anyhoo, the diode ring mixer produces a loss. The GC (Gilbert Cell) doesn't. In the transition from diode ring to the NE-602 there was a GC chip. I did a discrete build of one here.
Gilbert Cell from discretes
So the food for thought here is shall we build a GC from discretes, inset an RF amp to overcome the losses, or use an AF amp with some extra boost?