Get a hot air gun: it'll make your life way easier. You can tin the pads with a soldering iron, put the connector on and squirt some flux on the leads, and then just blow hot air until it reflows into place.
Sometimes though you just have to pile on solder and flux because the via is small enough that surface tension and heat dissipation means its never coming out
You can also scale this up in a solder oven and remove almost every single component. Used this for reversing a PCB a few times.
I have the SS-02, and I like it - I had one of the cheap blue ones first, but the pliable rubber tip really makes a difference. If you’re soldering smd by hand, it’s more than worth the $20
(If your foundry can't fabricate it, then make the pads thinner until they can fabricate the soldermask.)