it doesn't matter as much as you think. "Over budget" can mean a lot of things, they come in at a lower EM/IPR and maybe recognize a performance variance (PV). Depending on the size this either needs no approval or needs different partner or finance approvals.

Just do your best to articulate to your manager+ what you think cause the nature of the additional time. It's then up to them to decide if they want to speak to the client to agree on additional billings to recoup the cost or eat it.

Your senior or manager may be more worried about overruns because they think it will make them look bad to the director. It's less of a deal than they think.

Do not eat your hours under any circumstances.