How to make WP blog part of an existing website– without access to folders.

Hi all, new here.

My company wants to start a blog, but our current web designers are quoting too much to make one that’s integrated into our website for us (they estimate over 50 hours of work for $6500, does that seem reasonable?). I would just create a blog myself and upload WP into a sub-directory, but our CMS does not give us that ability. I can create a sub-directory and can edit the text/image content, but can’t deviate from their default layouts/settings. *more details at bottom*

So. I was wondering if there is a way around that. If I create a WP blog and host it on, say, Blue Host, can I direct the blog to show up at mainwebsite.com/blog or even blog.mainwebsite.com?

I hope all that made sense.

*If that wasn’t clear, here’s a more in depth description. When I log in to our control panel, I can create a subpage. They have pre-designed layouts for different kinds of pages (like a portfolio page vs a contact us page), so we choose from like 3 types. They ask for the page’s title and meta tags. Then, depending on what kind of page we chose, we can upload a thumbnail, a series of images with alt tags and blocks of text. The end. It’s like a less sophisticated version of the WP dashboard without the “editor” option, I guess.