If I solder these from the bottom pad, will the solder flow up through the hole and make a connection to the (more important) top pad?
Very unlikely, and if it doesn't you are stuffed.
Normally with home made boards you have a via just next to the pin. Too late for that now though.
I wouldn't do another board because there will probably be other bugs so you'll be doing another one anyway. What about doing this.
Drill the holes slightly over size if the pins are round (normal size if they are square), then get a very fine piece of wire (say a single strand from a multi-strand wire), solder that to the top track somewhere then poke it through the hole. Insert the terminal block and solder the bottom including the wire.
The block will sit up a fraction but not enough to worry you I wouldn't think.