Membrane keypads are mostly multiplexed, so you need to ensure that you're using the correct two connections for that specific button - depending on type though it may not be low enough resistance to make your meter beep.
The ones we have made for work are low enough, as they include nice mechanical dome switches - most of what I would call 'membrane' switches are low resistance, but not all.
Are you absolutely sure you checking on the correct 2 pins - it's often not very obvious.
Assuming you find the correct pins?, and assuming they are multiplexed?, then a small relay is probably the best and safest way to 'press' the button from a micro-controller. A CMOS switch 'might' work, but isn't guaranteed to, where a relay is.