"Delay" CSS files? Switch CSS instructions off?

My company has an ‘institutional site’. All pages are made into recognizable ‘company pages’ by means of a (sort of a) banner shown on top of all pages, with some company-wide links, a search box etc. It is compulsory that these things are shown on all pages. A number of company-wide CSS stylesheets are being used to shape this ‘top part’. This company site has nothing to do with Drupal, and I don't have any control over the company stylesheets.

Within the confines of my company site, I am running a Drupal subsite, with its own stylesheets. These CSS’s cover everything I need.

The problem, however, is that there are a number of ‘inheritances’ from the institutional stylesheets which I don’t want – and which, in fact, cause me major headaches, because a number of collisions occur. This, I think, is due to CSS classes which are used with the same name both in the institutional stylesheets as well as in my subsite stylesheets (e.g. body, #wrapper, #header etc.).

My question (maybe stupid, but hey). After the ‘top part’ of my pages has been formed by using the ‘institutional’ stylesheets, can I somehow ‘switch them off’ and have my own Drupal subsite stylesheets take control of everything which follows? So, basically, I want the Drupal site to “start” only after the company part has done its business. Is this possible? Or maybe put another way: can I ‘delay’ the theming process until the ‘institutional’ part is over? What would be the best approach to reconcile the company stylesheets with the Drupal stylesheets?