Hi, I have lots of experience soldering Lemo & Fischer connectors-but not clones. You must match wire size to the pin ID. 26 ga. should be OK with your 4 pin. You can always clip off a few strands if needed. I recommend using Teflon (PTFE stranded wire if possible as ordinary PVC insulation will melt. Any 40W to 75W iron is ok, but the tip has to be small & long enough to solder one contact at a time. It is a bit tricky, but I highly recommend using shrink tubing over each contact.
It you are using shielded cable, use a sharp dentist's pick to separate the strands, cut off enough of the strands to fit in the contact and slide a piece of shrink over the braid before soldering, leaving just enough unsoldered braid to fit in the contact. If your connector has crimp contacts this makes life much easier. I don't necessarily like crimping the contacts unless you have the right tool & wire, but it is much easier to solder the wire to the contact, slip a piece of shrink over the joint and then insert the wire/pin into the connector.
Hope this helps!