I've answered a couple of questions about Hanover before, here and here, if those are of interest.

I'm not sure what you mean by the British Crown meddling in German politics; the dukes of Hanover were also the monarchs of Britain, in the sense that they were distinct political entities who just happened to be sharing the same body. It would be difficult for the dukes of Hanover (technically electors and dukes of Brunswick-Lüneburg, but whatevs) to not involve themselves in German politics, and indeed, after Hanover was occupied during the Napoleonic wars, the Hanoverian foreign office was run out of London.

As to whether the dukes of Hanover could have been elected the HRE -- sure, it's not completely impossible, but the connection with Britain would likely have weighed against them; I very much doubt the other political entities involved in selecting the HRE would have wanted that connection.