If you insist
Selectively leaving parts of the profile publicly available: yes it's possible but not core. There's a lot of implications with custom development, so it'd need a very good use case to justify the development time, bug testing and potential impact on existing functionality.
The profile pages don't display much more information than the posting history and this is still available using the search function (instructions below). Outside this, it's mostly the profile posts (which is where the mod notes are sometimes posted), trading stats and personal information which members can choose to selectively display in the account settings.
You can easily find a banned member's posting history -
@Zwielicht mentioned using this --> site:blackhatworld.com "membername"
Alternatively, if you're already viewing one of their posts, you can bring up the others by clicking on the message:number at the top of the post:
Another way to do this is by using the search box. When you click in the text field, you'll see additional options that allow you to see all posts by a member (if you enter a username, you don't need to provide a keyword in the top search box)
You can also get the more detailed search page here which lets you narrow it down further:
https://www.blackhatworld.com/search/
Trading stats for banned members are still visible - just not through the profile page and getting to these is a little more complicated as you need the username and unique ID. You can get this by going to any of the banned member's posts - either in the search list (the results from doing the search above) or in the actual post itself. You need to right click on the link to the user's profile and select "copy link address". (I've screenshotted my profile postings list but you'll get the same link from any post or search result.)
Paste it in a text file, memorise it, screenshot it - whatever works best. You can see the member's trade by adding their username and ID to this url:
https://www.blackhatworld.com/trade/
For example, to see my trading profile you'd use this link:
https://www.blackhatworld.com/trade/apricot.655173/ - this works in the same way for any member, whether you're able to view their profile page or not.
If you're interested in seeing the information that's available, it's worth looking at the "Forums" "Trade" and "Members" dropdowns in the main menu bar on the top.
Bet you're wishing you hadn't dragged me in now...