hmm how about if you have the PC to send a "Are u ready?" signal to the uC before every new data transfer? If the uC is ready, it should reply and the PC should proceed with the data transfer. If the uC is doing some calculations, it should not reply at all, then the PC's uart would timeout.
That means you will need to enable uart only when the uC is not doing any calculations. When it is doing something, disable uart so that it does not get interrupted.