A.) Which Brinkmann?
B.) Porcelain coating on Weber Grill a B!t@h to mess with.
C.) You won't be able to use your Weber again.
I use my Weber fairly regularly, and I also have a Uniflame. Sometimes I use them to pre-burn logs because they smoke so much when starting. Especially before I made my drum smoker. I would look to find a drum, or an offset on Craigslist before I'd try to marry 2 such dis-similar units. Who knows--someone on here may be selling my--er--his offset unit. If he ever re-assembles it from painting the firebox.
On the other hand, if you put your mind to it, there's nothing stopping you. Look what Red Green builds!
Post pics! We love graphic visual aids! There are a couple different ways to add pictures. Just fiddle about, and modify the posts til it comes together.