Is it a bad idea to switch from using a quartz oscillator for my timing circuit to using a MEMS oscillator?
I've replaced a quartz oscillator module with a MEMS module and it was a straightforward transition. So the technology change should present no problems.

It's hard to see how the accuracy of the quartz oscillator won't be sufficient for the heart monitor, which are measuring a pattern less than 4 Hz. If you have more information to post on this, please do.

Otherwise, the crystal oscillator will be perfectly sufficient.