I wonder if it would be possible to simplify the task further, by removing cell reproduction processes? As far as I understand, an "immortal" non-replicating cell does not exist in nature, but there is nothing that prevents an artificial one from existing, especially if it has ways to repair cellular damage.
Many (if not most) end-of-lineage cell in higher mammals are non-replicating. They are, technically, not immortal, but some live a very long time, like neurons.
I like your experiment idea though, and I think it would be really interesting. If you publish your results, I can already see critics coming out of the bushes and shouting "immortality is immoral!", "unnatural!!!" and "it's a virus! it's a virus!". Oh how we like our value judgements...