Should I attempt to repair a leak in the radon mitigation system’s pipework

There are a variety of risks, including the risk that the system will fail to work as intended. You either punt those to let someone else design the system, or you take them on yourself when you design it.

To look at just one that you've thought of: Not growing bacteria in your water system is pretty much a matter of having clean water to start with and good sanitation when you work on the system. If your well is not contaminated, then sanitizing your water system whenever it's opened up for work should mean your system remains free of bacteria, if you don't introduce them. So, you probably need a filter (capable of excluding bacteria) on the air you'll be blowing through the packed media filter, for instance. Often, when you add up all the parts you need to do the job right, prebuilt systems don't look as expensive as they do when you haven't looked at the problem in depth; but that's not obvious until you do look into the details of the problem and solving it, reliably, completely, and without adding other problems.

If the air is not adequately filtered, you may need to add in a chlorination or other disinfection system after the filter to kill bacteria introduced in the filter.

Also - a packed media filter should reveal a few things to you in a search - it's more spraying the water over a physical medium that allows it to drip a lot than blowing bubbles. You need the booster pump because you lose all water pressure at this stage, so you pump from your well to the radon filter, and from the radon filter to your house. The thin layer of water on the media is more amenable to removing/exchanging gas than bubbles or droplets. The same type of system is often seen in waste treatment where a great deal of oxygen exchange is needed.

This from PSU extension has a pretty good discussion/description.

Here is a bit more on packed columns - you are interested in the process type called stripping or desorption.

This one is about removing VOCs and suffers a bit in translation but has good illustrations and removal of VOCs is the same basic process as removing radon.

First Radon is a pretty serious thing if you are working that much inside your basement. I have seen Radon levels in basement stay consistently at 4 times what is in the upper levels. Your 6.8 reading is high but not alarm bells until you get a few readings that high.

Initial steps:

• Open windows. Better than using fans is trying to create a convection pattern in your basement where the air is moving. Is your basement open enough to make this happen. Are your windows on sliders and easy to open. Get your basement aired out. Also don't close the doors by the steps. Let the basement air convect up to the upper level. Air flow is key.
• Seal any cracks that you have in your basement using caulk or epoxy sealants.
• Have your basement retested periodically over the next few weeks/months. I would say every 2-4 weeks.

If your future tests aren't dropping below the 4.0 pCi/L range than you need mitigation for sure. If they are falling below 4 then you need to figure out if you can keep up the good habits (airing the basement out continuously) even when the weather isn't great. So if your Radon levels drop with just improving airflow and sealing, then you need to test it every couple of months on its worst days. Radon levels can fluctuate greatly from day to day in a home (I had a house tested at 7.5 one day and two days later and me not touching airflow it was 4.3) so there is no way of knowing the "worst" day. But you can approximate during the winter "we haven't opened the windows in 3 days because it's cold as hell so let's test for Radon".

Now as far as mitigation, for most homes this will involve running a PVC pipe from your basement to the outside of your house - most of the time venting out your roof/attic. Sometimes the air cleaner is installed in the basement and sometimes in the attic. It probably involves opening a few walls. It is definitely a DIY type of job but that is a different question that would require a ton of details. I have installed Radon mitigation systems in basements in one day with just my son helping. A mitigation company will charge you anywhere between $800-2000 depending on the access to your basement floor (finished areas cost more), access to running the venting, how big your house is, if you have a circuit available to handle the electricity for the system, and what kind of system you want (you can get them at big box for $100-500 usually). DIY this could be a $200 job - it is just labor intensive - mainly patching drywall in your house.

How to get a good contractor? I would make sure they are certified to mitigate Radon. You can also call your local city hall and talk to the building inspector for local businesses that are certified or a suggestion. Radon mitigation is a lot of common sense (seal your basement and pump air out of it) so it doesn't take long talking to a contractor to tell if they understand how to do the job.