Does your company require you to   

"Does your company require you to take a flight if it means going over your 14 knowing you will time out before returning to base. We don't have much say in the matter. Forced overtime?"

That issue is moot. I can turn down any flight request.

Or- I can drop the med crew at receiving hospital and leg it back to base for relief within 14 hour duty.

Or leave your car keys at the base, relief pilot drives your car orrelief pilots' POV to receiving hospital to relieve you.

You have at least 2 hours to coordinate the relief process. Often the aircraft, crew were back at base before you got there.

Your duty period is to respond to all dispatch requests you can safely and legally accept. Sometimes the sending facility loses control and sometimes the patient goes suddem;y downhill.

It gets a little sticky when you can 'accept dispatch reasonably expecting to complete it within duty time limits' and the patient is FUBAR, eleventy-eight drips and vented, circling the drain, requiring extensive preparation. I've waited 8 hours before the crew loaded the patient before the medical crew had stabilized  the patient. Had one die onboard three times, twice landing at facilities en route for assistance.

It is what it is right then and there, deal with it, let the medical team do the medical thing and you just fly.