My memory may be worse than yours, but I the only tape that I remember something about is "foam tape". The one that is a mm thick. I think that "craft foam" from a hobby store would work better -- it is practically the same thing as foam tape, but without the sticky glue, and it comes in larger sheets.
The soldered pins are all on the underside of the PCB, and I don't think that it would be a good idea to put pressure on them. I think that some kind of spacers are needed that touch the circuit board in-between the pins. The spacers should be of a hard, insulating material. I am thinking that perhaps wood, like popsicle sticks, would work. Sheet plastic would be better, if you can get hold of some cheap. I have heard about "For Sale" signs being a cheap source of cheap sheet styrene in the US, but I have never seen it as I don't live there.
