If I file an amendment to my return, removing my father as a dependent while he lived with me, would he then automatically receive the $1,200 stimulus check?

No,  dependents do not get the stimulus money, & only dependents under 17 get it.  If you amend, amended returns must be mailed.  At this time the IRS is not processing mailed in returns.  They won't process mailed in returns until they re-open, so your amended return would not be processed in time to receive the stimulus money.

The stimulus money is actually an advance on your 2020 return (filed in 2021), so if he qualifies, he could receive it if he files a 2020 return.