Another common pattern would be to use explicit delegation (as opposed to inheritance, whose method dispatch can be considered a form of implicit delegation). I.e., as usual, put the stuff that remains after factoring out the commonalitities into separate modules that implement the same interface. Then have the base module delegate the respective functionality via method calls on whichever config/DB instance had been plugged in.

It somewhat depends on the context whether it's more appropriate to instantiate those delegates from within the base module, or from the main program, passing them to the constructor. For example, when using the latter approach, it would roughly look like this:

...
my $cfg;
if (whatever) {
    $cfg = Foo::Config->new(...);
} else {
    $cfg = Bar::Config->new(...);
}
my $obj = Base::Module->new( config => $cfg, ...);
...
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Generally, when things get complex, systems based on this simple form of delegation tend to remain more loosely coupled, with less interdependence among modules. In other words, they're often easier to modify in case the requirements should change.  At least that's my personal experience... but I'm pretty sure not everyone would agree :) — YMMV.