My How-To on this subject is now posted, here
Rufusbduck, it certainly doesn’t hurt to solder the middle pin, and I’m sure it helps at least a little with thermal transfer. I keep the pins straight which makes that middle pin a lot harder. The solder blob required to bridge the gap will often end up bridging across pins too.
Are you sure you use 400 F? That seems way too cold to melt solder properly. I’m up around 400C which is 750 F.
I also use 60/40 rosin core solder (diameter 0.025”, 0.6mm).