You need to consider mainly two factors:
1) When the project should start, if the project starts tomorrow, probably your only way is a custom ASP.NET MVC portal (or Django depends on the skills of the developers), if it can wait the release of Microsoft Portals Source Code, it should wait.
2) Don't focus on the integrations you have in the back, creating a convergence of the data inside a system (like put all the data before inside CRM) or inside an API (a REST endpoint that the web application will use to interact with the data) it's the "easy" part.
Focus on the capabilities the web application must have: assuming you need CRUD functionality (it's what we do over and over) which are the benefits of using Microsoft Portals instead of a custom web application and vice versa?
I know the ADX Studio/Microsoft Portal development fairly well, my primary job is not ASP.NET MVC developer but I had my fair share of hours creating web applications.
If you focus on the capabilities the selection on the platform will be easier, for example: your customer needs a CRUD on a custom entity (or on a common entity like account/contact) with search functionalities, make sure that the users can access only to their records? with Microsoft Portals you can configure this with very few hours, if you ask an ASP.NET MVC developer to create the same functionality (so index with a grid with sorting of the data, create page, edit page, delete page, search box, data security) it will take DAYS. And if we put on the plate for example a lookup field (like selecting the primary contact of the account) add more DAYS. You want to rearrange the fields or add/remove some fields, change the requirement level of the fields? add more DAYS. If you do these operations with a Microsoft Portals we talk about HOURS, not days, and requires a Portals customizer, not a developer.
The advantage of having the source code is that the project is an ASP.NET MVC and it can be extended (note: the ADX version 7 is a mix of Forms and MVC, nobody saw the source code of the new version yet) you want to create a custom page that shows data from Oracle and keeping the other pages from CRM? You can.
So, if the Microsoft Portals offers you most of the functionalities you need, than go with the Portals source code and deal with custom code for the rest part, if you need just a fraction of what a Portals can offer in term of CRUD, than maybe is better to go with a custom web application.
As I wrote before the integration of the systems is secondary, if you use the portals you can create (for example) an integration between the external system and a custom CRM entity, and expose this entity to the Portals, so you don't need to deal with connections to the external system at web application level, but only at CRM level.
Wrote this, working with the Microsoft Portals requires an intermediate/high level of training, but if you go with a custom web application you still need to find an intermediate/high level ASP.NET MVC developer, I saw too many developers saying that they know ASP.NET MVC and they only know how to do some scaffolding starting from the Visual Studio templates and they don't apply correctly the MVVM pattern, resulting in exposing confidential data or allowing malicious edits.
