The worst thing possible is mixed messaging, so what I should do is draw a clear line between Mobile Jazz "the services company" and Bugfender "the product". So each with their own marketing channels etc. However, to get traffic (potential customers) across I would think its a good idea to make some noise on the Mobile Jazz website (since it already is a strong brand) to get people across towards the Bugfender website to help build its own seperate brand. Also make sure that people can see that Bugfender is a product of Mobile Jazz (due to brand building), so I would make sure to include "Bugfender by Mobile Jazz" instead of just "Bugfender" in all my messaging. At least until it can stand on its own two feet.

My two cents. Hope it helps :)