There have been several threads on related issues; I don't know if this is all of them or not. Pik(s)* vs. piek(s)* vs. other words seems to be a contentious issue, and possibly partly regional, as well as possibly a matter of age or register, and/or traditionalist vs. descriptivist.
I agree that 'shot' AE / 'jab' BE is the most common way to express the concept of getting vaccinated, whether as an initial shot or a booster. Those words are used in many cases where German would use impfen/Impfung, including in journalism where a short word is useful in headlines.
Especially in the current context of Covid, just 'booster' alone is also usually enough, without other words, though 'booster shot' is also common.
It seems right to caution that 'booster vaccine shot / booster vaccine dose' (OP) is a mouthful, too many words together -- too formal or medical, too much repeated description, not as idiomatic as other expressions.
'Vaccine(s)' and 'dose(s)' are probably used more in public-health contexts, like discussing the logistics of which drug to provide and how to get enough doses to enough people. I wouldn't use those in an individual context, like discussing whether you have gotten your third shot or when you can schedule it, or discussing getting shots for children.
For a more formal register, 'vaccination' is better when referring to an individual shot going into an arm.