I have a full basement in a cold climate (North Dakota). The previous owner had a raw hole broken out of the cement floor that must have served as an emergency sump pump hole. We haven't had water in 8 years and did landscaping around the house to slope water away from house. Due to Radon issues, we filled and sealed the hole and finished the installed a subfloor with Dry Core in the event of moisture in the basement. This year we have a really high water table and water is in the basement - had to rip up carpet, etc. Now I am wondering if I should break through the floor again to install a sump pump. A contractor said he could do an outdoor sump pump for us, but I do not have an electrical source nearby, which will raise the costs and headache. I am also uncertain if the cold climate will require me to remove the pump every year or what kind of hassles are associated with an outdoor sump pump. What would you do? Start over in the basement or put in an outdoor pump. My Radon mitigation system is also a worry if we install a pump inside the house-- not sure how this will effect the deal.
I wouldn't even consider one or the other... both.
An outdoor sump pump in our could climate (I'm in Minnesota) is a real pain in the neck. A good friend of mine has one in his old house and he has to bring it in every winter and put it back out when the snow starts melting, which often means a few hours digging to find the top of the sump pit, then every couple days checking and adjusting the position of the pump as the solid ice on the bottom of the pit melts. Worse was the year it melted, then re-froze with the pump in it... I recall he had a number of very unpleasant things to say about it the next day at work. ;)
Depending on your levels of radon, the mitigation could be pretty simple. Several of my co-workers have a small bathroom exhaust fan in the utility room where there sump is located that ducts out of the house. It's on a timer that runs it for X minutes every Y hours, depending on what the levels were in their house. Most of us agreed that that was better than having to deal with flooded basements every few years.
I'm not understanding the connection between the two issues. They really are different systems. As for Radon, not sure if it varies from state to state, but in MN they come in drill holes in your basement slab, then run PVC pipes from that up and out your roof with a low-power fan continually venting the system. The goals is to put negative pressure on your slab so that Radon doesn't enter your space. Since it sounds like you want a finished basement, I'd definitely go that route.
As for the water issue, if you have a high water table, maybe finishing the basement isn't the best idea. One thing I did with our old house is I didn't put down any flooring material and instead acid-stained the concrete. Not a look everyone likes, but I'll never have to deal with tearing out wet carpet. ;)
If your basement is flooding, and you live in freezing climates, an indoor sump pump is a pretty important. A high water table also means a finished basement is going to be difficult.
If you are installing the sump pump to deal with the water table, and you also need to mitigate Radon, look at a sump pump basin something like this. It is sealed from above so Radon gas won't come into the house, but still has holes for the Radon vent pipe and the water discharge.
Note that you will want to install the sump basin in a location that works well for both the Radon vent pipe and the water discharge. (This may not be where the previous owner's sump hole was installed before.)
Also, note that this style of sump basin won't give you an open pit that you can easily dump water into, because it's sealed on top. For example, you can't run an overflow hose from your water softener over to the sump pit. If you put open holes in the pit, you will undo the seal that blocks the Radon. (You might be able to attach a drain line with a P-trap, but the radon vent pipe suction would probably empty the trap over time.)
Not sure about your state, but in a lot of places, Radon work has to done by a licensed contractor. So, it would probably be a good idea to talk to a few Radon contractors and explain your sitution (which is a little more complex than a typical Radon system).
If you haven't researched Radon mitigation systems yet, you really should do so.