Should I show overbudget amount on fixed fee invoice?

I suppose it depends on whether the cost overruns can be demonstrated to be related to incorrect scoping (not yours) or additional requests by the client, whether the work is recurring, what the agreement provides for, etc. Also, depends on what the norm is in your industry.

An over-simplified response perhaps but if the agreement doesn't have any provision for recovery of the overruns I don't know that there is added benefit to highlighting it on an invoice vs. doing a more general post-mortem on the project which might help the client understand how you would price it differently in the future.

Interested in what others have say on the topic.

Fixed fee is fixed fee. Invoice the fixed fee. One way you can do this is you can accompany the invoice with a scope of work you did (that shows the costs you incurred).

Let me turn it around.... if you overestimated and profited handsomely, would you show it on the invoice?

I agree with Emerson that a fixed fee is a fixed fee. In my opinion, showing the overage on the invoice would be like complaining that the job didn't turn out as planned. That would not help your relationship with your client.

I think the more important thing, is that you determine why there was a cost overrun. If you underestimated the cost then you'll need to improve your estimating process. If you just spent more than the budget, you'll want to improve your project management skills. If you provided services outside the scope of service you'll want to improve your system for identifying when this is about to happen and creating a change order for the additional cost.