We use a term here "Don't put lipstick on a pig!" I understand your challenge and concern with your reputation, but that is the beauty of projects...we are working in a world of unknowns and you don't know what you don't know. Look at this as an opportunity!
Look back at how you came to the budget....
Are the cost overruns coming from a missing party in those discussions? If so, go to the person and understand what their spend has been to date, as what they need to complete and ask for where they can cut to still achieve quality and deadlines.
If you didn't miss anyone or their savings aren't going to cut it, look for low hanging fruit. Is there anything that was added in recently. You could go back to the team and say "I loved that idea and think it would add to the product, but we need to reduce some costs here and focus on meeting the requirements, not over delivering."
Negotiations and big spenders are your next step. Traditionally, your overruns come from people and a missed calculation on how much time it would take them. Go back and ask them why the extra time? Anything to reduce that? Do they need all level 3 engineers or could they use a lower level resource? (What's the risk to time and quality with that choice?) If it is in service or parts, you could go back to your vendor and negotiate. The life-cycle of your project is often where they make their money...they are interested in this launching also.
Challenge - Your people should have been the main source of input and they told you it would cost x, take this many hours, etc. Challenge their overruns. Push back and ask them to fix it...bring back some alternatives to rain this thing in.
Lastly, put the picture together and present it to your project sponsor. Let him/her know that you are currently looking at an overrun...here are some reasons, this is what I have done to bring it under control, and here are the options going forward. A) Gets us under budget, but time gets pushed out B) Only 5% more money, but we launch on time...etc etc.