We are staying one night at La Familia hotel, and need to get to Gare Lyon in the morning. It has been our experience in the past that many taxis are not interested in taking you such a short distance. On the other hand it's difficult navigating the metro with suitcases, changing levels, and connections. In this case more than one stop.

Anyone have a good suggestion for getting there less stress?

Your hotel will call a cab for you. Hotels can always get cabs for their guests. Shouldn't be a problem. I've always been told to ask in the morning before departure as taxis arrive within 10 minutes of calling. However to be safe check with the hotel the night before.

Call Taxi Bleu or have your hotel call them night before....no hassle, cheap price....

I agree with you about the short distance and taxi attitudes, but in this situation, I think it's far enough that there wouldn't be a problem, even if the hotel weren't calling it. But they are. That may be farther than you are thinking, mappy.fr says it is 2.6 km (they also claim you can drive that in 4 min which is impossible, even if the traffic on the quais and the bridge isn't heavy, which it may be.

and unless things have changed the meter will start running when the taxi wheels start rolling

Dudkey is right, expect X euros already on the meter because the flag goes down when the taxi receives the call from your hotel. But, you will have no problem booking a taxi