My recommendation would be to leave that one where it is.What would you recommend for aftercare?
I have the opportunity to dig up a Kalamata olive tree that's quite old. The base is around 60-80cm in diameter.
A tree that size will require a large hole to get under the lignotuber so be prepared for a lot of digging. The lignotuber is typically larger under the ground so the base could be twice that size? You'll need to dig a hole at least 3 times the size of the lignotuber to get access to roots around and under the stump and that's a lot of soil to remove - and then replace back in the hole after getting it out.
My guess is the stump will weigh in at around 50-80kg, maybe a lot more, so even getting it out of the hole will be a challenge unless you have a machine or some of those mates with plenty of muscles but fewer brains. After you get it home a container will be a challenge - be aware that all collected trees seem to expand as they come out of the hole. They are always bigger than they seem initially. Allow for a container at least half as big again as you initially estimated.
Some pics of an olive dug in 2015 by a couple of our younger members and it is only around 40 cm diameter lignotuber. That one took an hour to dig and both of them to lift out of the hole. If the tree survives and becomes presentable as a bonsai you'll need a pot. Any idea how big that pot will be? Price of said pot?
The reality is that you are probably young and over optimistic so you'll go ahead with this crazy plan.
Use potting mix for any plant in a container. It is possible to maintain plants in garden soil but there's all sorts of complications so not recommended.
I usually place collected olives in part shade until I get good growth indication some new roots but I've planted collected olives direct into the grow beds in full sun and they all survived. Just make sure it does not totally dry out for the first year or so.
Some olives seem to recover better than others. Some collected olives were strong after just 1 year, some take several years to grow good new roots.