The hanger bolts you've linked are primarily for attaching something with holes to a wood item. You've got the right idea but should be considering the inversion of the machine threaded portion.
Look for threaded inserts for wood. You would have to drill a reasonably larger hole in the wood, force the insert into the hole (usually with a screwdriver, sometimes a hex key) and then thread your tap handle into the insert.
The above image is listed as an insert for hard wood. Depending on how tightly the grain lines are on your live edge, you may want to use an insert for soft wood:
By appearance, I'd guess that the threads on the tap are 1/4-20 and the inserts linked above are available in that thread type. You can take your tap to the local hardware store to match up the thread if you are not certain. You may also find that the big box stores and hardware stores will carry these products.
You should also discover that the product will provide the necessary hole size to be drilled in order to properly use the insert. I've also discovered that the hardwood inserts require to be tapped with a standard tap, an additional expense for your project. It appears from the product information that the softwood inserts can be directly driven. If the wood is not soft enough, the insert will probably be rendered useless.