Yes, I agree with you that Past Perfect should be used.
But, as an auxiliary verb 'have' is the marker of the Perfect aspect.
Past perfect of have is 'had'.
The past tense of 'get' is 'got. The past perfect of get is 'had got'.
If you just use 'got' you would be using the past tense and not the past perfect tense.
This is long-winded but I hope that this explanation helps.