"Took over from the German police, so said Baptiste. dodgy therefore".
Not necessarily. Jurisdiction is governed by thev Status of Forces Agreement (which is backed by international agreements/law)
www.nato.int/cps/en/natohq/official_texts_17265.htm
When there is is the possibility that the crime is committed by someone other than stationed forces (and the crime is a crime in both UK and Germany, and the matter does not related to serving personnel on their official duties) then it would be investigated by the German authorities.
And there was a scene in ep.1 where Brig offered personnel in support of the German investigation. So a certain amount of jointery, but not lead in the investigation. So German police would end their investigation into the death once satisfied it was suicide of Alice, but continued the investigation and prosecution of the suspected abductor, because they had primacy.
So the only potentially hooky bit is the DNA test. Because the DNA repository for the samples of military personnel is in UK, not Germany, and it's a store of actual samples that will only be sequenced should there be otherwise unidentifiable remains to be identified. Indeed the consent for holding such a sample (which is voluntary, but recommended) is specific that that can be the only possible use.
To request it in support of a German investigation into the death of a dependent takes a shit load of paperwork in the first place, and the sequencing is done in UK. There is no British military capacity to do that in Germany, so it's not clear who ran the sample from the burnt corpse - because that has to be either a German lab, or a sample sent back to UK. And either of those scenarios involves far too many people who would notice interference with either samples or subsequent paperwork.
I think that it is a clumsy point, which was not well researched (no obvious way to fake any part of test/results). So it would be a pity if any great weight of plot was attached to it.
As posted above, I think Baptiste's comment was more a creation of a brooding atmosphere of pervasive corruption. And he didn't know yet that 'Alice' was still alive and that the body in the shed is not the girl everyone thought was sleeping in it.