What kind of bike is it ? On many Harley's the signal housing is a full unit replacement with wires already coming out. You must have obtained a new or different signal bullet from another source, but still why can't you use the wires that are already in the housing and just splice them in the harness OR I guess they aren't long enough to get back to the connectors in the neck and you don't want to splice them. But if you heat sink the splice wiith a black colored heat shrink it should be ok.
If you have your bulb socket popped out you should be able to see how to melt the solder. Can you post a pic ?