Though, here's another idea. Could he persuade them he's meeting some girl? Would any of the young ladies of the Jr Auxiliary be willing to play along? He could tell Rabastan Lestrange he's spending time with his new girlfriend and wants a bit of privacy, that seems like something even a Death Eater might understand. It would be a reason he might come to Doughty Conduit on a near-daily basis.

The trouble is that we have to assume they're watching, even if they give Harry space. But perhaps we could use polyjuice and a volunteer provide a facsimile of Harry, to drink tea and hold hands with some facsimile of Miss Perks or Miss Bones. Might prove a bit awkward back at school, though, I suppose...