+1 for thin strand. I would unravel some desoldering wick and use one or two strands of that.
No lead free solder within 100 miles of this project!
Using a vinyl cutting machine (or an X-acto knife and a cutting board) you can make a little mask that leaves only the part of the gold traces exposed that you want to solder, with a slight clearance (think: crude solder mask). This is to prevent solder going "too far" but also to help contain the flux to the traces only, instead of spreading to unwanted areas.
Conservative tip heat setting, enough to melt the solder and not much more. Pre-heat with conservatively hot air.
Paste flux, minimal quantity (speck on end of a toothpick) to avoid it spreading/seeping to somewhere it's not wanted, as cleaning this module may be a delicate affair.
Tin the traces, then add the (lightly fluxed) thin strand(s) without adding more solder.