Plugged vias are not a panacea for via in pad. The plugs can expand and outgas during soldering. For anything small that I care about, I either use VIPPO or I don't put vias in the pad. It solved many problems and I have never looked back.
If you are doing a one-off and don't want the expense of VIPPO, putting a weight on may be kludgy, but if it's a one-off (or just a few), who cares?
John
Hi All, just in case anybody finds this thread, I managed to get the chips soldered properly.
I got new boards. I decided to go with the ~40% cost increase of a 4 layer, thicker board so that they can plug and overplate my vias. That was a mistake to assume I can kludge it at home. Another change is the solderpaste. I got Loctite GC 10 and it's really nice, it prints a whole lot better too and is apparently a lot less picky about temperature.
Thanks for everyone for chiming in and giving tips!