Oh, I think I was misunderstood. I meant to feed the 3.7V directly into the tablet, not the battery. In fact, no battery would be involved except for the external one: feed its output through your 3.7V regulator into the contacts inside your tablet, bypassing the internal battery (in fact, keep it somewhere else). (Yes, I realise it's still a risky stupid idea :-) but no battery charging would be involved.

Though I don't know what the third terminal of the battery does, or how the tablet uses it. Current sensor? Temperature sensor?