What you're looking for is a condensate pump, and there are a dozens on the market ranging from $50 junk price up to $400. the tube runs down to the condensate trap and pumps it where ever you want it to go.

"We'd really like to get better AC to the bedrooms, as the heat pump in the living room just doesn't reach there very well."

The problem, as stated, suggests a problem moving conditioned air where it needs to go. Assuming the heat pump is correctly sized for the total cooling load and the duct work is properly designed, fixing the problem may be as simple as installing balancing dampers in the duct work. By adjusting the balancing dampers, air flow can be redistributed--decreasing flow to the Living Room and increasing air flow to the bedrooms. If you can't do that, another option to consider is installation of a thermostat with a remote sensor in one or more of the bedrooms. The thermostat can be programmed to control based on input from the remote sensor(s) at night. Yes, that option will over-cool some areas of the home to improve the comfort level in the bedrooms, but the cost penalty shouldn't be too great in an 850 SF +/- home.

The damper adjustment option is the way without unnecessary and wasteful spending. If you have airflow to the bedrooms look at better ways to get to that point. Add dampers. Or reconfigure runs. Limit what you are asking a hvac person to only those options when you getup a service call.

A condensate pump is a solid solution for routing drainage to a bathroom trap

I’ve used one myself

When I was learning the basics, the info on HVAC training cost gave me a clear picture of what to expect

good point from Charles Ross Homes — condensate pump solves drainage, but if the real issue is airflow distribution, that's a separate fix. worth getting both diagnosed together before spending on either

an independent HVAC assessment could sort out whether this is a simple damper fix or needs something bigger

Thanks for all your comments. There is no duct work - the current unit is wall mounted in the living roomm and the fan just blows the air into the room - it only gets to the other rooms by pushing it with fans. I'm thinking maybe the condensate pump makes sense if I can drain it into the bath drain. I'll look at those.