Yes, you can take heat from the bathtub and heat the cup of water with it, but you will have to do work in the process. For example, you can run a heat pump that moves heat from the tub to the cup, but the heat pump will need to be powered by electricity, or maybe a hamster on a wheel.
The size of the bathtub and the cup of coffee are immaterial as long as they both contain a large number of atoms. Temperature is an intensive quantity, one that does not change based on the absolute size of the system. That means that the statement "heat flows naturally from high temperature to low temperature" does not depend on the size of the reservoirs involved.
In this case, you are moving heat from 283K to 293K. To move one Joule of heat across that temperature gradient requires $1 - \frac{283}{293} = \frac{10}{293}$ Joules of work at minimum. This is for a theoretical ideal heat engine. To do it in real life, you will need more work than that because you cannot be perfectly efficient.
Note: I suppose there are cute responses to this question which would refer to waiting for times longer than the age of the universe or asking for random fluctuations smaller than could possibly be measured. Here, I'm trying to give a practical response.