Not that slow a leak, I would say, and think the leak is probably more urgent than getting the bath repainted!
To answer your question, can you contrive a way of diverting the water directly into the waste outlet? For example, by putting a length of wastepipe to run directly from the dripping tap (faucet) to the waste.
I would advise you sort the drip out though, as otherwise you'll end up with limescale all over your new bath surface.