To make the case for bespoke development. There are a lot of benefits, your directly attacking the problem you have, directly and precisely. You are not using a hammer as your only tool sometimes you need a spanner.
Agility – Its lighter and possibly more agile as your development can be adapted to suit your goals as they change and develop.
Speed and efficiency – you can spend a lot of time bending an existing package to your will. To the point at which you almost break it.
Faster – often your not carrying a pile of *features* which you never wanted but got as a side effect of the package you adapted.
Easily adapted – more easily adapted with the kind of features that come up client demands, specific legislation or business focus changes.
Intuitive – usage can be designed to flow in a way that fits with existing systems of thinking and terminology so it becomes a lot more intuitive.
Unique to you – hopefully your organisation does something unique, your competitors can also use off the shelf software so are you removing a possible advantage over them by using the same.
Directed – Start with what you want to do and go from there no compromises.
Licensing – you own it a bespoke development is generally going to avoid on going licensing costs that may be associated with existing packages.
So for your untypical problems and challenges perhaps you need a untypical and bespoke solution.