Web forwarding/masking confusion

This is a bit similar to other questions regarding masking and forwarding, but not exactly the same (as far as I can tell). So, apologies if I’ve overlooked something…

I have a (wordpress) website, www.my_main_site.com, hosted by company A, which is up-and-running and all fine.

I’d like to send out a marketing newsletter, which will be a link to a sort of mini-website, (a non-wordpress site) hosted by company B.

However, I’d like the url to the mini-website to be www.my_main_site.com/my_birthday

Can this be done, or would I need to move both sites under the umbrella of a single host?

I suppose an alternative is to simply create an iframe on the main site? Maybe that’s the way to go? … I’ll look into how to create an iframe on WordPress…