Running Craft alongside another CMS on the same domain
You're asking a loaded question. I don't think there's a wrong answer here but if it were me, you have a couple options.
Addressing the Elephant in the Room
How good of a salesman are you? ;) You mention the e-commerce CMS isn't getting the job done but replacing it is off the table. I've gone down this road before.  Since you've got your team interested in Craft, why not consider Craft Commerce too? I know what you're thinking. It's a tough sell. But then again, so is integrating 2 CMS and making sure things don't break between them. What happens if that 3rd party CMS vendor doesn't &quot;approve&quot; of what you're doing and cuts support?  Not saying it'll happen but like you mentioned... politics has a way of messing with people.
I don't know how much business logic that the current e-commerce solution has that you'd have to replicate or how many products and variations you're dealing with but if anything is up for the challenge, it's Craft and Craft Commerce. Just demoing the seamless integration between the two might be enough for the sales/marketing team. Time is money for them.
If you can't go with Craft Commerce, it really depends on what the other software does and how you integrate the two.
Integrating the Two
I'm assuming both CMS will live on the same physical server? Indeed, Craft could  play alongside the other CMS. That's the easiest option. If it were me, I'd probably setup all new marketing pages within Craft and then link to the actual ordering pages in 3rd party CMS.
By default, most CMS including Craft are going to suck up any request that isn't a static file, query the database, and then return the &quot;virtual&quot; page from the database. The actual page doesn't really exist on the server, so Craft looks for &quot;would-be&quot; 404s. That's basically what the rewrite rules in a .htaccess file does—it tells Apache to send any 404 back to the application (usually index.php).
Why is this important? If you have any flexibility with the 3rd party CMS, I'd &quot;quarantine&quot; it to a folder on the server, like /store/ so any request within that folder (like /store/black-leather-jacket/) would go to the commerce CMS (assuming the CMS is sane and doesn't do something stupid like have absolute URLs).
Craft would handle everything else. You could quarantine &quot;Craft&quot; to a folder, but like you mentioned, any page you create would have a URL like /marketing/black-leather-jacket/demo/ or whatever you decide. That's sub-optimal in my opinion. As far as SEO and flexibility go, you probably want people to hit your marketing/landing pages first. People won't buy if they don't have info.
(And it's probably easier to do a 301 redirect to the already existing pages... you could setup Craft so that if it can't find a file, have it take a look if the page exists in the other CMS and redirect if it does.)
Tapping into the Data
Assuming the e-commerce database design/schema is easy to tap into, I'd write a Craft plugin to grab product data directly from the database, based on whatever parameters you need to feed a Twig template. That's beyond the scope of this question but the dev is relatively straightforward.
Craft is built on Yii, an excellent PHP framework so whatever Yii can do Craft can do as well. Craft also ships with Guzzle which would make it easy to &quot;scrape&quot; the actual page or as Matt P mentions building or tapping into some sort of bridge between the 2 systems.
Speaking of scraping, if the 3rd party e-commmerce solution is a hosted solution, then you might have to set that one up on another domain and let Craft handle the marketing pages. In this case, you might consider something like store.domain.com.
Again, all comes down to what the other e-commerce solution does. Craft is relatively flexible any way you slice it.