I've tried Frost one and 3M and both work well. ALso, the "Toolstation" zinc spray works well too and is about 1/2 the price of the Frost can. Its sold as a zinc primer, but when sprayed and dried is silver, much like the Frost one (some zinc primers such as the Screwfix one are grey and dont work too well).
The Toolstation one says on the can that it is spotweldable. I use these primers on the inside of butt welded repairs when closing a box section or between plug welded overlaps. I find the welding works better if you clean the accesible bits to bare metal once the repair piece is clamped up.
I use the Bilt Hamber "Etchweld" which is very good but not sure if available in US. The only US auto supplier I know of is Eastwood who do supply weldable primer and as already suggested Upol is available - but personally I don't really like it, not the easiest to weld through (maybe just me)
I suggest you visit your local auto paint supplier or bodyshop and see what they suggest. Maybe some of our American friends on this site will have better suggestions for products over the pond.
Yes, that would do the job. I would still remove, with wire brush or sandpaper any paint on the accessible side of the weld on the car body. And get rid of all the rust as well.