Two different questions. Toilet paper "explosions" and sewer smells point to problems in your septic tank/cesspool system, particularly the venting thereof. A sump pump for handling "clean" water coming into the basement is a different matter.

It's not hard to come up with a way to divert the water to the outside of the house. The real question is one of how you can utilize that water effectively. Sump pumps traditionally use a 1 1/4" discharge line to allow a high volume of water to be pumped in a very short timeframe. These pumps don't put out very much pressure because pressure isn't needed for sump pit applications. All that is required of a sump pump is that it lifts water up to eight feet vertically so that it can gravity-flow to a ditch or storm sewer line. You need to keep that in mind when figuring out how you will pipe this water to your gardens. This pump will push the water to the exterior of your home but the water pipes outside need to be large diameter and run horizontally or slightly downward. Your other option would be to have a holding tank that the sump pump dumps into and then use a traditional jet pump/pressure tank system to pressurize the water for your irrigation system. And you thought this would be easy, right?

"My plumber is familiar with water-powered pumps, but says they're not compatible with a sealed sump..." Find another plumber. Add a vent to a sealed sump and then it is exactly the same as a non-sealed sump.

It sounds like you answered your own question: you have a sump that needs almost constant pumping, so you definitely should get the most reliable system you can afford. You won't go wrong with a Home-Guard. They are essentially bullet proof, and will pump forever. I have no experience with Baseguard, but if it is sized right (i.e., has the ability to pump the maximum amount you will ever need) it will probably work fine, as these types of pumps have no moving parts and are generally very reliable. And, as you have painfully learned, when a pump runs as much as your's, you need to check on it and clean it out at least monthly, if not weekly.