You need to change your gridColumnconfigure and gridRowconfigures to use the '-minsize' attribute. '-weight' just controls how the cells respond to the grid being resized. As an example:

for (my $i = 0; $i < $rows; $i++) {
    $tl->gridRowconfigure($i, -weight => 1, -minsize => 80);
}
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would make your cells always at least 80 pixels wide. All you have to do is make sure that minsize is greather than or equal to the largest widget.

I have used this (many times!) in TCL/Tk, but not in Perl/Tk. However, I can't see why it wouldn't work.