The important thing is your customer facing person(for that matter your manager) should be aware that even if he or she thinks the bug is trivial and can be fixed in an hour always stick to the 2 weeks target. You can use the same bug as the reason why you don't want to rush the fix early. Tell the customer that you are not happy with the fact that your development process introduced this bug and don't want to repeat the same in the bug fix also and promise to deliver a high quality fix in two weeks time.
In my experience most of the customers are happy to know that the problem will not be repeated than the fix for current problem because they might already have found a workaround to move their business forward. Obviously they cannot stop all the business and wait for your fix.
By the way I was working in a company offering SaS and the customers daily business was dependent on the product. The customers used to have high expectation on the turn around time.
Their idea of an offer you can't refuse is an offer... and you'd better not refuse.