If the config isn't going to change very often I would just throw it in S3 and have each lambda load it at startup.
If it does change a lot or you want to build some sort of UI to update the settings then you could go the DB route. If you load config when the lambda starts up - ie outside of the handler function you would only need to load the config once as long as the lambda doesn't go back to sleep so the penalty probably isn't that steep.
// this will only be loaded when the Lambda starts up
// keep in mind if you are loading from an external resource
// it will probably be async so your function should return a Promise
var config = someFunctionThatLoadsConfigFromS3();
// entry point for Lambda will be called for every event(api gateway request)
module.exports.handler = function(event, context) {
config.then(function(config) {
// do some stuff with config
context.done(null, 'return a value');
}).catch(....);
};