Yeah, 5 regular FM stations fit into 1MHz, but they are spaced further apart to avoid interference in cheap radios. If you used a narrow-band IF the 1MHz drift would include about 100 narrow-band stations. It would appear to make the drift 20 times worse.
I was mistaken before, adding a dual conversion mixer and new 455KHz IF would require you to turn-down your transmitter's input so the volume would end up the same, BUT, the radio would be much more sensitive to weak signals 'cause it would have more IF amplification.
I looked into Google at the TDA1083 IC that is in your radio. It is a Telefunken "one chip AM-FM radio and speaker amplifier". Its datasheet shows two radio circuits, one with many extra parts that seems to be not very good, but not bad. The other radio circuit is very simple and its performance is, uh, you know.
He, he which circuit do you think is in your radio? If it is a Telefunken radio, it probably "won't be very good, but not bad". It's not a Telefunken, is it.
Look how much range NASA and now the europeans get with their space probes. I know how huge their receiving dishes are, but they must have RF amplifiers and IF strips coming out of their wing-wang! I bet their transmitters are pretty good too. Of course they use very accurate crystals in both.